Mathematical Modeling of Double Filtration by Colloidal Theory with Study Water Containing Microcystis spp.

Hataishi, Laís Ayumi; Botari, Alexandre

doi:10.1590/1678-4324-2023220860

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Article - Engineering, Technology and Techniques • Braz. arch. biol. technol. 66 • 2023 • https://doi.org/10.1590/1678-4324-2023220860 copy

Mathematical Modeling of Double Filtration by Colloidal Theory with Study Water Containing Microcystis spp.

Authorship SCIMAGO INSTITUTIONS RANKINGS

Abstract

Anthropic activities have been causing serious impacts in aquatic environments, deteriorating the quality of the waters. Superficial springs with excess of nutrients, rich in compounds of phosphor and nitrogen, may induce blooms of algae and cyanobacteria, hard to be removed in the treatment of water. Such microscopic particles, e.g. Microcystis spp., are named biocolloids. This work propose to apply the mathematical modelling by the extended colloidal XDLVO theory (Extended Derjarguin-Landau-Verwey-Overbeek) for coagulation and gravel upflow filtration (GUF) and sand downflow rapid filtration (SDRF) to the technology of double filtration applied to study water containing Microcystis spp. The XDLVO theory has been shown to be efficient at evaluating the behavior of the colloidal particles as a function of the separation distance, making apparent how the intermolecular and surface forces act: acid-base interaction (AB), Lifshitz-Vander Walls (vdW), double electric layer (DEL) and Born repulsion forces. Such forces that act in this colloidal system formed by Microcystis spp. were analyzed in terms of their mutual interaction and their interaction with porous environments in double filtration at the stable thermodynamic situations (pre-coagulation) and at unstabilized by the chemical coagulation. The raw water energetic barrier of repulsion and after the sand downflow rapid filtration gave an average decreasing of about 90% in absolute values.

Keywords:
cyanobacteria; biocolloids; XDLVO theory.

HIGHLIGHTS

• The application of mathematical models allows the study of complex environments.

• The chemical coagulant acts on electrostatic destabilization, decreasing the absolute value of the electrical potential of the electrical double layer.

• The XDLVO theory is efficient in studying the behavior of energies and interaction forces along the particle separation distance.

Parameters	Essay 2
pH	6.64
Apparent color (uH)	24
True color (uH)	2.0
Turbidity (uT)	2.36
Total alkalinity (mg CaCO₃/L)	6.0
Total hardness (mg CaCO₃/L)	20
Acidity	3.0

Hour (h)	T (ºC)	pH	ZP (mV) SW	pH	ZP (mV) CW	pH	ZP (mV) GUF	pH	ZP (mV) SDRF
1	22.50	7.02	-28.8	6.92	-26.6	6.74	-2.00	6.68	-1.80
3	21.75*	7.09*	-34.6	6.97	-24.0	6.88	-1.20	6.90*	-2.00
6	21.75*	7.24	-26.1	6.96	-25.2	6.86	-2.50	6.88	-2.10
7	21.00	7.09	-31.3	6.96	-21.4	6.86	-1.92*	6.90*	-1.98*
10	21.75*	7.01	-24.9	6.96	-21.7	7.09	-2.00	7.13	-2.00
Partial Efficiency			-		-		46.2%		0.3%
Total Efficiency			-		-		99.7%		99.9%

Equation	References
$* Δ G^{D E L} = π . ε_{a} . r_{p} . [2 φ_{p} φ_{c} . l n (\frac{1 + exp (- K δ)}{1 - exp (- K δ)}) + (φ_{p}^{2} + φ_{c}^{2}) . \ln (1 - exp (- 2 K δ)]$	[²⁹29 Wu H, Shen F, Wang J, Wan Y. Membrane fouling in vacum membrane distillation for ionic liquid recycling: Interation energy analysis with the XDLVO approach. J Memb Sci. 2018;550:436-47.]
$* Δ F^{D E L} = 64 π . ε_{a} . K . r_{p} . {(\frac{k_{B} . T}{Z . e})}^{2} . \tanh (\frac{Z . e . φ_{p}}{4 k_{B} . T}) . \tanh (\frac{Z . e . φ_{c}}{4 k_{B} . T}) . \exp (- K . δ)$	[²⁶26 Nikoo AH, Kalantariasl A, Malayeri MR. Propensity of gypsum precipitation using surfasse energy approach. J Mol Liq. 2020;300(112320):1-15.]
$* * Δ G^{D E L} = 2 π . ε_{a} . r_{p} . φ_{p}^{2} [\ln (\frac{1 + \exp (- K δ)}{1 - \exp (- K δ)}) + \ln (1 - \exp (- 2 K δ)]$	[²¹21 Zhang T, Zhan J, Wang Q, Zhang H, Wang Z, Wu Z. Evaluating of the performance of natural mineral vermiculite modified PVDF membrane for oil/water separation by membrane fouling model and XDLVO theory. J Memb Sci. 2022;641(119886):1-12.]
$* * Δ F^{D E L} = 32 π . ε_{a} . K . r_{p} . {(\frac{k_{B} . T}{Z . e})}^{2} . \tanh {(\frac{Z . e . φ_{p}}{4 k_{B} . T})}^{2} . \exp (- K . δ)$	[³⁰30 Ou Q, Xu Y, Li X, He Q, Liu C, Zhou X, et al. Interactions between activated sludge extracelular polymeric substances and model carrier surfaces in WWTPs: A combination of QCM-D, AFM and XDLVO prediction. Chemosphere. 2020;253(126720):1-10.]
$K = \sqrt{\frac{2 F^{2} . I .10 ³}{ε_{a} . R . T}}$	[²⁶26 Nikoo AH, Kalantariasl A, Malayeri MR. Propensity of gypsum precipitation using surfasse energy approach. J Mol Liq. 2020;300(112320):1-15.]
$* Δ G^{A B} = 2 π . r_{p} . λ_{A B} . Δ G_{δ_{0}}^{A B} . \exp (\frac{δ_{0} - δ}{λ_{A B}})$	[²¹21 Zhang T, Zhan J, Wang Q, Zhang H, Wang Z, Wu Z. Evaluating of the performance of natural mineral vermiculite modified PVDF membrane for oil/water separation by membrane fouling model and XDLVO theory. J Memb Sci. 2022;641(119886):1-12., ²⁵25 Fan Z, Ji P, Zhang J, Segets D, Chen D, Chen S. Wavelet neural network modeling for the retention efficiency of sub-15 nm nanoparticles in ultrafiltration under small particle top ore diameter ratio. J Memb Sci. 2021;635(119503):1-11., ³⁰30 Ou Q, Xu Y, Li X, He Q, Liu C, Zhou X, et al. Interactions between activated sludge extracelular polymeric substances and model carrier surfaces in WWTPs: A combination of QCM-D, AFM and XDLVO prediction. Chemosphere. 2020;253(126720):1-10., ³¹31 Zou W, Zhao J, Sun C. Adsorption of Anionic Polyacrylamide onto Coal and Kaolinite Calculated from the Extended DLVO Theory Using the van Oss-Chaudhury-Good Theory. J Polym. 2018;10(113):1-11.]
$* Δ F^{A B} = 2 π . r_{p} . Δ G_{δ_{0}}^{A B} . \exp (\frac{δ_{0} - δ}{λ_{A B}})$	[²⁸28 Botari A, Di Bernardo L, Dantas AD. Análise de modelação matemática da interação entre partículas na filtração direta utilizando a teoria coloidal. Eng Sanit Ambient. 2012;17(1):81-94.]
$* * Δ G^{A B} = π . r_{p} . λ_{A B} . Δ G^{A B} . \exp (\frac{δ_{0} - δ}{λ_{A B}})$	[²⁸28 Botari A, Di Bernardo L, Dantas AD. Análise de modelação matemática da interação entre partículas na filtração direta utilizando a teoria coloidal. Eng Sanit Ambient. 2012;17(1):81-94.]
$* * Δ F^{A B} = π . r_{p} . Δ G^{A B} . \exp (\frac{δ_{0} - δ}{λ_{A B}})$	[²⁸28 Botari A, Di Bernardo L, Dantas AD. Análise de modelação matemática da interação entre partículas na filtração direta utilizando a teoria coloidal. Eng Sanit Ambient. 2012;17(1):81-94.]
$Δ G_{δ_{0}, p w c}^{A B} = 2. [\sqrt{γ_{3}^{+} .} (\sqrt{γ_{1}^{-}} + \sqrt{γ_{2}^{-}} - \sqrt{γ_{3}^{-}}) + \sqrt{γ_{3}^{-} .} (\sqrt{γ_{1}^{+}} + \sqrt{γ_{2}^{+}} - \sqrt{γ_{3}^{+}}) - \sqrt{γ_{1}^{+} . γ_{2}^{-}} - \sqrt{γ_{1}^{-} . γ_{2}^{+}}]$	[³²32 Flores AG, Solong SK, Heyes GW, Ilyas S, Kim H. Bubble-particle interactions with hydrodynamics, XDLVO theory, and surfasse roughness for flotation in na agitated tank using CFD simulations. Miner Eng. 2020;152(106368):1-11.]
$Δ G_{δ_{0}, p w}^{A B} = - 4. (\sqrt{γ_{2}^{+}} - \sqrt{γ_{3}^{+}}) . (\sqrt{γ_{2}^{-}} - \sqrt{γ_{3}^{-}})$	[³³33 Yang L, Wen J. Can DLVO theory be applied to MOF in diferente dieletric solventes? Microporous and Mesoporous Materials. Microporous Mesoporous Mater. 2022;343(112166):1-9.]
$* Δ G^{v d W} = \frac{H . r_{p}}{6 δ} . [1 - \frac{5,32 δ}{λ_{w}} . \ln (1 + \frac{λ_{w}}{5,32 δ})]$	[²⁰20 Yang Y, Yuan W, Hou J, You Z. Review on physical and chemical factors affecting fines migration in porous media. Water Res. 2022;214(118172):1-16.]
$* Δ F^{v d W} = \frac{H . r_{p}}{6 δ ²} . {(\frac{5,32 δ}{λ_{w}} + 1)}^{- 1}$	[²⁰20 Yang Y, Yuan W, Hou J, You Z. Review on physical and chemical factors affecting fines migration in porous media. Water Res. 2022;214(118172):1-16.]
$* * Δ G^{v d W} = \frac{H . r_{p}}{12 δ} . [1 - \frac{5,32 δ}{λ_{w}} . \ln (1 + \frac{λ_{w}}{5,32 δ})]$	[²⁸28 Botari A, Di Bernardo L, Dantas AD. Análise de modelação matemática da interação entre partículas na filtração direta utilizando a teoria coloidal. Eng Sanit Ambient. 2012;17(1):81-94.]
$* * Δ F^{v d W} = \frac{H . r_{p}}{12 δ ²} . {(\frac{5,32 δ}{λ_{w}} + 1)}^{- 1}$	[²⁸28 Botari A, Di Bernardo L, Dantas AD. Análise de modelação matemática da interação entre partículas na filtração direta utilizando a teoria coloidal. Eng Sanit Ambient. 2012;17(1):81-94.]
$H_{p w c} = (H_{12}^{1 / 2} - H_{33}^{1 / 2}) . (H_{22}^{1 / 2} - H_{33}^{1 / 2})$	[³⁴34 Xie Q, Saeedi A, Piane CD, Esteban L, Brady PV. Fines migration during CO2 injection: Experimental results interpreted using surfasse forces. Int. J. Greenh. Gas Control. 2018;65:32-9.]
$H_{p w} = (H_{12}^{1 / 2} - H_{33}^{1 / 2}) ²$	[²⁶26 Nikoo AH, Kalantariasl A, Malayeri MR. Propensity of gypsum precipitation using surfasse energy approach. J Mol Liq. 2020;300(112320):1-15.]
$* Δ G^{B o r n} = - \frac{H . x^{6}}{7560} . [\frac{8 r_{p} + δ}{{(2 r_{p} + δ)}^{7}} + \frac{6 r_{p} - δ}{δ^{7}}]$	[³⁵35 Lee H, Kang S, Kim SC, Pui DY. Modeling transport of coloidal particles through polydisperse fibrous membrane filters undes unfavorable chemical and physical conditions. Powder Technol. 2019;355:7-17., ³⁶36 Mahmoudi D, Rezaei M, Ashjari J, Salehghamari E, Jazaei F, Babakhani P. Impacts of stratigraphic heterogeneity and release pathway on the transporto f bacterial cells in porous media. Sci Total Environ. 2020;729(138804):1-11., ³⁷37 Muneer R, Hashmet MR, Pourafshary P. Fine Migration Control in Sandstones: Surface Force Analysis and Application of DLVO Theory. ACS Omega. 2020;5:31624-39.]
$* Δ F^{B o r n} = - \frac{H . x^{6} . r_{p}}{180 δ^{8}}$	[³⁸38 Ruckenstein E, Prieve DC. Adsorption and desorption of particles and their chromatographic separation. AIChE J. 1976;22(2):276-83.]
$* * Δ G^{B o r n} = - \frac{H . x^{6} . r_{p}}{1260. δ^{7}}$	[²⁶26 Nikoo AH, Kalantariasl A, Malayeri MR. Propensity of gypsum precipitation using surfasse energy approach. J Mol Liq. 2020;300(112320):1-15.]
$* * Δ F^{B o r n} = - \frac{H . x^{6} . r_{p}}{180. δ^{8}}$	[²⁰20 Yang Y, Yuan W, Hou J, You Z. Review on physical and chemical factors affecting fines migration in porous media. Water Res. 2022;214(118172):1-16.]

Parameters	Definition	Values [Units]	References
$k_{B}$	Boltzmann constant	1.3808.10^-23 J.K^-1	[³⁹39 Yang Z, Hou J, Pan Z, Wu M, Zhang M, Wu J, et al. A innovative stepwise strategy using magnetic FE3O4-co-graft tannin/polyethyleneimine composites in a coupled process of sulfate radical-advanced oxidation processes to control harmful algal blooms. J Hazard Mater. 2022;439(129485):1-11., ⁴⁰40 Xia T, Li S, Wang H, Guo C, Liu C, Liu A, et al. Insights into the transport of pristine and photoaged graphene oxide-hematite nanohybrids in saturated porous media: Impacts of XDLVO interactions and surface roughness. J Hazard Mater. 2021;419(126488):1-11.]
$e$	Electron charge	1.602.10^-19 C	[³⁹39 Yang Z, Hou J, Pan Z, Wu M, Zhang M, Wu J, et al. A innovative stepwise strategy using magnetic FE3O4-co-graft tannin/polyethyleneimine composites in a coupled process of sulfate radical-advanced oxidation processes to control harmful algal blooms. J Hazard Mater. 2022;439(129485):1-11.]
$ε_{a}$	Permissiveness coefficient of the solution	6.9539.10^-10 C²/Jm	[³⁹39 Yang Z, Hou J, Pan Z, Wu M, Zhang M, Wu J, et al. A innovative stepwise strategy using magnetic FE3O4-co-graft tannin/polyethyleneimine composites in a coupled process of sulfate radical-advanced oxidation processes to control harmful algal blooms. J Hazard Mater. 2022;439(129485):1-11., ⁴⁰40 Xia T, Li S, Wang H, Guo C, Liu C, Liu A, et al. Insights into the transport of pristine and photoaged graphene oxide-hematite nanohybrids in saturated porous media: Impacts of XDLVO interactions and surface roughness. J Hazard Mater. 2021;419(126488):1-11.]
$R$	Avogadro's constant	6.02.10²³ mol^-1	[³⁹39 Yang Z, Hou J, Pan Z, Wu M, Zhang M, Wu J, et al. A innovative stepwise strategy using magnetic FE3O4-co-graft tannin/polyethyleneimine composites in a coupled process of sulfate radical-advanced oxidation processes to control harmful algal blooms. J Hazard Mater. 2022;439(129485):1-11.]
$F$	Faraday's constant	96485 C/mol
$r_{p}$	Particle radius	0.000002 m	[³⁹39 Yang Z, Hou J, Pan Z, Wu M, Zhang M, Wu J, et al. A innovative stepwise strategy using magnetic FE3O4-co-graft tannin/polyethyleneimine composites in a coupled process of sulfate radical-advanced oxidation processes to control harmful algal blooms. J Hazard Mater. 2022;439(129485):1-11., ⁴¹41 Li W, Qi W, Chen J, Zhou W, Li Y, Sun Y, et al. Effective removal of fluorescente microparticles as Cryptosporidium parvum surrogates in drinking water treatment by metallic membrane. J Memb Sci. 2020;594(117434):1-10.]
$φ_{p} / φ_{c}$	Particle/Collector zeta potential *	V	[¹⁵15 Kuroda EK. Remoção de células e subprodutos de Microcystis spp. por dupla filtração, oxidação e adsorção [doctoral thesis]. São Carlos: Escola de Engenharia de São Paulo, Universidade de São Paulo; 2006. 267 p.]
$H$	Hamaker's constant*	J	[²⁸28 Botari A, Di Bernardo L, Dantas AD. Análise de modelação matemática da interação entre partículas na filtração direta utilizando a teoria coloidal. Eng Sanit Ambient. 2012;17(1):81-94., ⁴²42 Xu H, Jiang H, Yu G, Yang L. Towards understanding the role of extracelular polymeric substances in cyanobacterial Microcystis aggregation and mucilaginous bloom formation. Chemosphere. 2014;117:815-22.]
$γ$ ⁺/ $γ$ ^-	Recipient/Donor surface tension*	mJ/m²	[²⁸28 Botari A, Di Bernardo L, Dantas AD. Análise de modelação matemática da interação entre partículas na filtração direta utilizando a teoria coloidal. Eng Sanit Ambient. 2012;17(1):81-94., ⁴³43 Kwon B, Park N, Cho J. Effect of algae on fouling and efficiency of UF membranes. Desalination. 2005;179:203-14.]
$T$	Temperature*	K	[¹⁵15 Kuroda EK. Remoção de células e subprodutos de Microcystis spp. por dupla filtração, oxidação e adsorção [doctoral thesis]. São Carlos: Escola de Engenharia de São Paulo, Universidade de São Paulo; 2006. 267 p.]
$Z$	Valence of the ion	1 (admensional)
$λ_{w}$	Wavelength of light scattered between particles	100 nm	[⁴⁴44 Yang Y, Hou J, Wang P, Wang C, Wang X, You G. Influence of extrecellular polymeric substances on cell-NPs heteroaggregation process and toxicity of cerium dioxide NPs to Microcystis aeruginosa. Environmental Pollution. 2018;242:1206-16.]
$K$	Debye-Huckel constant*	m^-1	[²⁵25 Fan Z, Ji P, Zhang J, Segets D, Chen D, Chen S. Wavelet neural network modeling for the retention efficiency of sub-15 nm nanoparticles in ultrafiltration under small particle top ore diameter ratio. J Memb Sci. 2021;635(119503):1-11.]
$X$	Collision diameter	0.5 nm
$λ_{A B}$	Wavelength of scattered light for acid-base interaction	0.6 nm
$δ$	Separation distance	nm
$δ_{0}$	Minimum separation distance	0.16 nm	[⁴⁵45 Ren B, Weitzel KA, Duan X, Nadagouda MN, Dionysiou DD. A comprehensive review on algae removal and control by coagulation-based processes: mechanism, material, and application. Sep Purif Technol. 2022;293(121106):1-26.]
$I$	Ionic strength of the system	0.0212 mol/L	[⁴4 Zhang B, Tang H, Huang D, Gao X, Zhang B, Shen Y, et al. Effect of pH on anionic polyacrylamide adhesion: New insights into membrane fouling based on XDLVO analysis. J Mol Liq. 2020;320(114463):1-9.]

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[1] *Correspondence: laishataishi@gmail.com; Tel.: +55-44-99709-6757 (L.A.H.)

TREATMENT PHASES	MAXIMUMS		MINIMUMS
TREATMENT PHASES	Value (kT) Distance (nm)	Value (nN) Distance (nm)	Value (kT) Distance (nm)	Value (nN) Distance (nm)
SW (S-S)	224854.78	210.00	-2.33.10¹²	-6.65.10¹²
SW (S-S)	0.16	0.29	0.01	0.01
CW (S-S)	157352.03	209.12	-2.33.10¹²	-6.65.10¹²
CW (S-S)	0.17	0.29	0.01	0.01
GUF (S-S)	30359.15	174.22	-2.33.10¹²	-6.65.10¹²
GUF (S-S)	0.20	0.29	0.01	0.01
GUF (S-P)	19138.79	111.47	-3.26.10¹¹	-9.29.10¹¹
GUF (S-P)	0.17	0.25	0.01	0.01
SDRF (S-S)	30367.70	174.24	-2.33.10¹²	-6.65.10¹²
SDRF (S-S)	0.20	0.29	0.01	0.01
SDRF (S-P)	19218.23	111.68	-3.26.10¹¹	-9.29.10¹¹
SDRF (S-P)	0.17	0.25	0.01	0.01