Novo modelo empírico de viscosidade para suspensões compostas utilizadas em experimentos de fluxos gravitacionais de sedimentos

Castro, Camila; de Oliveira Borges, Ana Luiza; Manica, Rafael

Camila Castro

Universidade Federal do Rio Grande do Sul, Porto Alegre, RS, Brasil

https://orcid.org/0000-0002-0481-0815

Ana Luiza de Oliveira Borges

Universidade Federal do Rio Grande do Sul, Porto Alegre, RS, Brasil

https://orcid.org/0000-0001-5613-6837

Rafael Manica

Universidade Federal do Rio Grande do Sul, Porto Alegre, RS, Brasil

https://orcid.org/0000-0002-0527-1374

E-mails: camila.castro@ufrgs.br (CC), alborges@iph.ufrgs.br (ALOB), manica@iph.ufrgs.br (RM)

Camila Castro: Run the experiments, analysis and wrote the manuscript. Ana Luiza de Oliveira Borges: Was Camila Castro supervisor on the master of science study and review the text. Rafael Manica: Was co-supervisor of Camila Castro and review the text and discussions.

Editor-in-Chief: Adilson Pinheiro Associated Editor: Iran Eduardo Lima Neto

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Figure 1
Rheological diagram (shear stress and shear rate relationship) for all mixtures.

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Figure 2
Apparent viscosity related with shear rate for all Herschel-Bulkley models obtained in the experiments.

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Figure 3
Experimental data of the relative viscosity of the mixtures as a function of volumetric concentration and the comparison with seven equations proposed in the literature. The mixtures composed by: (A) 100% kaolin; (B) 85% kaolin; (C) 75% kaolin; (D) 50% kaolin.

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Figure 4
Experimental data of the relative viscosity of the mixtures as a function of volumetric concentration and the comparison with seven equations proposed in literature: (A) 25% kaolin; (B) 15% kaolin; (C) 0% kaolin.

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Figure 5
(A) Relation between the relative viscosity(μ_r) and volumetric concentration (ϕ) for all Newtonian flows; (B) Logarithm of the relative viscosity as a function of parameter ϕ/ϕ_max for mixtures with non-Newtonian behavior.

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Figure 6
(A) Relative viscosity of the mixture suspensions proposed in this work and the rheological threshold between Newtonian and Non-Newtonian behavior. The dashed line represents a threshold variation of ±15%; (B) Rheological threshold for three mixtures composed of cohesive and non-cohesive material. Kaolin and coal mixtures (red dotted line), present work; kaolin and sand mixtures (Coussot & Piau, 1994Coussot, P., & Piau, J. K. (1994). On the behavior of fine mud suspensions. Rheologica Acta, 73(3), 175-184. http://dx.doi.org/10.1007/BF00437302.
http://dx.doi.org/10.1007/BF00437302... ; Kiryu, 2006Kiryu, H. S. (2006). Investigação reológica e análise mecânica de compósitos não-Newtonianos (Dissertação de mestrado). Faculdade de Engenharia de Ilha Solteira, Universidade Estadual Paulista, Ilha Solteira.; Santos 2003Santos, F. L. (2003). Estudo teórico-experimental em via de determinação de lei de atrito em escoamentos de fluidos hiperconcentrados (Dissertação de mestrado). Faculdade de Engenharia de Ilha Solteira, Universidade Estadual Paulista, Ilha Solteira.) (blue traced line) and kaolin and glass beads (black line) (Manica, 2009Manica, R. (2009). Geração de correntes de turbidez de alta densidade: condicionantes hidráulicos e deposicionais (Tese de doutorado). Program in Water Resources and Environmental Sanitation, Universidade Federal do Rio Grande do Sul, Porto Alegre.).

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Table 1
Rheological models classification. The variables are: shear stress (τ) and shear rate (

\dot{γ})

apparent viscosity (μ_ap) and yield strength (τ₀); k and n are power law coefficients.

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Table 2
Usual relative viscosity equations found in the literature.

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Table 3
Relative viscosity (μ_r) experimental data at a shear rate of 25 s^-1. The gray background indicates a Herschel-Bulkley models and the white a Newtonian Model.

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Table 4
Relative Viscosity error between the experimental results and seven models from literature.

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Table 5
Values of intercept (α), gradient (β) and coefficient of determination (R²) of the linearized equations of .

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Table 6
Maximum packing fraction comparison between this work (mixture kaolin and coal) and Manica (2009)Manica, R. (2009). Geração de correntes de turbidez de alta densidade: condicionantes hidráulicos e deposicionais (Tese de doutorado). Program in Water Resources and Environmental Sanitation, Universidade Federal do Rio Grande do Sul, Porto Alegre. mixture kaolin and glass beads).

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(15)

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(18)

(19)

(20)

(21)

(22)

Type	Rheological models		Yield strength $(τ_{0})$	Eq.
	Bingham	$τ = τ_{0} + μ_{a p} \dot{γ}$	Yes	(1)
Non-Newtonian	Herschel Bulkley	$τ = τ_{0} + k {\dot{γ}}^{n}$	Yes	(2)
	Pseudo-plastic (n<1)	$τ = k {\dot{γ}}^{n}$	No	(3)
	Dilatant (n>1)	$τ = k {\dot{γ}}^{n}$	No	(3)
Newtonian	Newton	$τ = μ \dot{γ}$	no	(4)

Author	Equation	Limit	Eq.
Einstein (1905)Einstein, A. (1905). Berichtigung zu meiner Arbeit: Eine neue Bestimmung der Moleküldimensionen. Annalen der Physik, 34, 591-592.	$μ_{r} = 1 + 2.5 ϕ$	ϕ < 2.5%	(5)
Maron & Pierce (1956)Maron, S. H., & Pierce, P. E. (1956). Application of ree-eyring generalized flow theory to suspensions of spherical particles. Journal of Colloid Science, 11(1), 80-95. http://dx.doi.org/10.1016/0095-8522(56)90023-X. http://dx.doi.org/10.1016/0095-8522(56)9...	$μ_{r} = {(1 - \frac{ϕ}{ϕ_{m a x}})}^{- 2}$	ϕ >20%	(6)
Dougherty & Krieger (1959)Dougherty, T. J., & Krieger, I. M. (1959). Potential around a charged colloidal sphere. Journal of Physical Chemistry, 63(11), 1869-1872. http://dx.doi.org/10.1021/j150581a019. http://dx.doi.org/10.1021/j150581a019...	$μ_{r} = {(1 - \frac{ϕ}{ϕ_{m a x}})}^{- [μ] ϕ_{m a x}}$	ϕ >40%	(7)
Chong et al. (1971)Chong, J. S., Christiansen, E. B., & Baer, A. D. (1971). Rheology of concentrated suspensions. Journal of Applied Polymer Science, 15(8), 2007-2021. http://dx.doi.org/10.1002/app.1971.070150818. http://dx.doi.org/10.1002/app.1971.07015...	$μ_{r} = {[1 + \frac{7,5}{(\frac{ϕ_{m a x}}{ϕ} - 1)}]}^{2}$	ϕ <45%	(8)
Batchelor & Green (1972)Batchelor, G. K., & Green, J. T. (1972). Determination of the bulk stress in a suspension of spherical particles to order c2. Journal of Fluid Mechanics, (56), 375-400.	$μ_{r} = 1 + 2.5 ϕ + k ϕ^{2}$	2 < ϕ < 10	(9)
Manica (2009)Manica, R. (2009). Geração de correntes de turbidez de alta densidade: condicionantes hidráulicos e deposicionais (Tese de doutorado). Program in Water Resources and Environmental Sanitation, Universidade Federal do Rio Grande do Sul, Porto Alegre.	$μ_{r} = 1 + ϕ (2.24 + 0.44 % c l a y)$	2 < ϕ < 35	(10)
Santamaría-Holek & Mendoza (2010)Santamaría-Holek, I., & Mendoza, C. I. (2010). The rheology of concentrated suspensions of arbitrarily-shaped particles. Journal of Colloid and Interface Science, 346(1), 118-126. PMid:20303498. http://dx.doi.org/10.1016/j.jcis.2010.02.033. http://dx.doi.org/10.1016/j.jcis.2010.02...	$μ_{r} = (1 - ϕ_{e f}) - [μ_{i}]$ and		(11)
	$φ_{e f} = \frac{ϕ}{1 - [(1 - ϕ_{m a x}) ϕ / ϕ_{m a x}]}$		(12)
Brouwers (2010)Brouwers, H. J. H. (2010). Viscosity of a concentrated suspension of rigid monosized particles. Physical Review E, 81(5), 051402. PMid:20866225. http://dx.doi.org/10.1103/PhysRevE.81.051402. http://dx.doi.org/10.1103/PhysRevE.81.05...	$μ_{r} = {(\frac{1 - ϕ}{1 - (ϕ / ϕ_{m a x})})}^{(μ ϕ_{m a x}) / (1 - ϕ_{m a x})}$		(13)
Horri et al. (2011)Horri, B. A., Ranganathan, P., Selomulya, C., & Wang, H. (2011). A new empirical viscosity model for ceramic suspensions. Chemical Engineering Science, 66(12), 2798-2806. http://dx.doi.org/10.1016/j.ces.2011.03.040. http://dx.doi.org/10.1016/j.ces.2011.03....	$μ_{r} = 1 + μ ϕ + z ϕ {(\frac{ϕ}{ϕ_{m a x} - ϕ})}^{2}$	ϕ <40%	(13)

Mixture		Maximum packing fraction	Volumetric concentration ϕ (%)
%Kaolin	%Coal	(ϕ_max)	5	10	15	20	25	30
100	0	0.33	2.53	4.27	20.69	60.59	233.3	756.5
85	15	0.35	2.22	3.26	12.99	41.23	127.9	362.5
75	25	0.38	2.12	2.91	4.09	35.61	70.4	173.3
50	50	0.40	1.75	2.42	3.33	11.77	19.8	47.3
25	75	0.37	1.63	2.12	2.69	3.54	8.8	24.9
15	85	0.40	1.34	1.93	2.48	3.30	5.1	11.0
0	100	0.50	1.31	1.81	2.06	2.82	3.5	7.3

Mixture		Einstein	M&P	Chong	B&G	S, H &M	Browers	Horri
% Kaolin		Relative Error (%)
		Equation 5	Equation 6	Equation 8	Equation 9	Equation 11	Equation 12	Equation 13
100	Mean	86	76	32	86	84	85	56
	St. Dev.	19	21	25	19	18	19	29
85	Mean	83	69	27	82	79	80	50
	St. Dev.	22	25	18	22	22	22	12
75	Mean	77	62	46	76	73	74	32
	St. Dev.	23	28	53	23	23	23	24
50	Mean	70	45	16	68	62	64	33
	St. Dev.	24	22	15	24	22	23	11
25	Mean	59	24	10	56	37	45	30
	St. Dev.	20	7	10	19	8	12	4
15	Mean	51	17	8	47	29	34	23
	St. Dev.	20	14	7	18	8	10	10
0	Mean	42	8	10	37	25	25	15
	St. Dev.	15	7	10	12	6	6	9
Totals	Max	83	69	46	82	79	80	50
	Min	42	8	8	37	25	25	15
	Mean	67	43	21	64	56	58	34
	St. Dev.	17	27	14	18	25	24	14

*%Clay*	α	β	R²
100	1488.7	4.62	0.9840
85	540.4	4.25	0.9883
75	382.2	3.87	0.9812
50	113.7	3.39	0.9604
25	57.5	4.79	0.9909
15	23.5	2.94	0.9495
0	-	-	-

Mixture	Maximum packing fraction (ϕ_max)
%Kaolin	This work	Manica (2009) Manica, R. (2009). Geração de correntes de turbidez de alta densidade: condicionantes hidráulicos e deposicionais (Tese de doutorado). Program in Water Resources and Environmental Sanitation, Universidade Federal do Rio Grande do Sul, Porto Alegre.
100	0.33	0.32
85	0.35	0.34
75	0.38	0.37
50	0.40	0.48
25	0.37	0.60
15	0.40	0.58
0	0.50	0.62

[1] E-mails: camila.castro@ufrgs.br (CC), alborges@iph.ufrgs.br (ALOB), manica@iph.ufrgs.br (RM)

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Novo modelo empírico de viscosidade para suspensões compostas utilizadas em experimentos de fluxos gravitacionais de sedimentos

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