Liquid-phase RTD studies in a flotation column: an analysis between tanks-in-series and axial dispersion models

Couto, Hudson Jean Bianquini; Marques, Carlos Henrique da Fonseca; Braga, Paulo Fernando Almeida

doi:10.1590/0370-44672019740107

Hudson Jean Bianquini Couto ⁴E-mails : 4 hcouto@cetem.gov.br, hudson.couto@ifrj.edu.br

Centro de Tecnologia Mineral - CETEM, Mineral Processing and Technologies Coordination, Rio de Janeiro - Rio de Janeiro - Brasil.

Instituto Federal do Rio de Janeiro - IFRJ, Rio de Janeiro - Rio de Janeiro - Brasil.

http://orcid.org/0000-0002-1812-4162

Carlos Henrique da Fonseca Marques ⁵ 5 caiquefm@poli.ufrj.br

Universidade Federal do Rio de Janeiro - UFRJ, Escola Politécnica, Departmento de Metalurgia e Materiais, Rio de Janeiro- Rio de Janeiro - Brasil.

http://orcid.org/0000-0002-7769-0201

Paulo Fernando Almeida Braga ⁶ 6 pbraga@cetem.gov.br

Centro de Tecnologia Mineral - CETEM, Mineral Processing and Technologies Coordination, Rio de Janeiro - Rio de Janeiro - Brasil.

http://orcid.org/0000-0001-9093-746X

E-mails : 4 hcouto@cetem.gov.br, hudson.couto@ifrj.edu.br

5 caiquefm@poli.ufrj.br

6 pbraga@cetem.gov.br

Cell	Tracer (phase)	Variable	Model	Reference
Column (D_c= 0.075 m, Hc=4.57 m)	KCl (L)	J_g, J_f	PFR	*Rice et al.* (1974)**RICE, R. G.; OLIVER, A. D.; NEWMAN, J. P.; WILES, R. J. Reduced dispersion using baffles in column flotation. Powder Technology,v. 10, n. 4, p. 201-210, 1974.
Column (D_c=0.06 m, Hc=1.1 m)	HCl (L)	-	PFR	*Ityokumbul et al. (1988)*ITYOKUMBUL, M. T.; KOSARIC, N.; BULANI, W. Parameter estimation with simplified boundary conditions. Chemical Engineering Science, v. 43, n. 9, p. 2457-2462, 1988.
Columns (D_c= 0.46; 0.91 m, H_c=9.5;10 m)	fluorescein (L), MnO₂ (S)	D_c, D_p	PFR	Dobby and Finch (1985)DOBBY, G. S.; FINCH, J. A. Mixing characteristics of industrial flotation columns. Chemical Engineering Science, v. 40, n. 7, p. 1061-1068, 1985.
Column (D_c= 0.08 m, H_c=1.0 m, V=4.4 L)	KCl (L)	J_g, J_f	PFR_r, CSTR_s	*Mavros et al. (1989)*MAVROS, P.; LAZARIDIS, N. K.; MATIS, K. A. A Study and Modelling of Liquid-Phase Mixing in a Flotation Column. International Journal of Mineral Processing, v. 26, p. 1-16, 1989.
Column (D_c=0.054 m, H_c=1.55 m, V=3.55 L)	NaCl (L), ⁸⁹Au₁₉₁ (S)	J_g, %S, D_c, D_b	PFR_r, CSTR_s	Mills and O'Connor (1990)MILLS, P. J. T.; O'CONNOR, C. T. The modelling of liquid and solids mixing in a flotation column. Minerals Engineering, v. 3, n. 6, p. 567-576, 1990.
Column (D_c= 0.054 m, H_c=2.3 m)	⁸⁹Au₁₉₁ (S)	J_g, %S, Q_f, C_f, H_f, J_w	PFR_r, CSTR_s	Goodall and O'Connor (1991)GOODALL, C. M.; O'CONNOR, C. T. Residence time distribution studies in a flotation column. Part l: The modelling of residence time distributions in a laboratory column flotation cell. International Journal of Mineral Processing, v. 31, p. 97-113, 1991.
Column (D_c= 0.102 m, H_c=4.0 m)	KCl (L)	J_g	PFR	Xu and Finch (1991)XU, M.; FINCH, J. A. The axial dispersion model in flotation column studies. Minerals Engineering, v. 4, p. 553-562, 1991.
Columns (D_c= 0.1; 2.5 m, H_c=3;13 m)	Conductivity (L)	J_g	PFR	Finch and Dobby (1991)FINCH, J. A.; DOBBY, G. S. Column flotation: a selected review. Part I. International Journal of Mineral Processing,v. 33, n. 1, p. 343-354, 1991.
Columns: lab (D_c= 0.05 m, H_c=1.9 m) and industrials (D_c= 1.2 e 0,91 m, H_c=11 m)	Ammonium and potassium bromide - Br-82 (L,S)	D_c, D_p	PFR, CSTR_s	*Mills et al. (1992)*MILLS, P. J. T.; O'CONNOR, C.T., The mixing characteristics of solid and liquid phases in a flotation column. Minerals Engineering, v. 5, n. 10-12, p. 1195-1205, 1992.
Column (D_c=0.054 m, H_c=2.3 m)	⁸⁹Au₁₉₁ (S)	J_g, %S, Q_f, C_f, H_f, J_w	PFR_r, CSTR_s	Goodall and O'Connor (1992)GOODALL, C. M.; O'CONNOR, C. T. Residence time distribution studies in a flotation column. Part 2: the relationship between solids residence time distribution and metallurgical performance. International Journal of Mineral Processing, v. 36, p. 219-228, 1992.
Columns	(L,S)	D_p	PFR	Xu and Finch (1992)
Column (D_c= 0.05 m, H_c=1 m)	KCl (L,S)	J_g, J_f, D_c	PFR	*Mankosa et al. (1992)*MANKOSA, M. J.; LUTTRELL, G. H.; ADEL, G. T.; YOON, R. H., A study of axial mixing in column flotation. International Journal of Mineral Processing, v. 35, n. 1, p. 51-64, 1992.
Columns (D_c =(0.025; 0.04; 0.08 and 0.11 m)	KCl (L)	J_g, J_f, H_L	PFR	Mavros and Daniilidou (1993)MAVROS, P.; DANIILIDOU, A. Mixing in flotation columns. Part II. Liquid-phase residence time distributions studies. Minerals Engineering, v. 6, n. 7, p. 707-719, 1993.
Column (D_c=0.06 m, H_c=1.1 m)	HCl (L)	J_g	PFR	*Ityokumbul et al. (1994)*ITYOKUMBUL, M.T.; KOSARIC, N.; BULANI, W. Gas hold-up and liquid mixing at low and intermediate gas velocities I. Air-water system. Chemical Engineering Journal, v. 53, n.3, p. 167-172, 1994.
Column (D_c=0.06 m, H_c=1.1 m)	HCl (L)	J_g, C_f, %S	PFR	*Ityokumbul et al. (1995)*ITYOKUMBUL, M. T.; KOSARIC, N.; BULANI, W. Effect of fine solids and frother on gas hold-up and liquid mixing in a flotation column. Minerals Engineering, v. 8, n. 11, p. 1369-1380, 1995.
Industrial Mechanical (V=148 m³ and 160 m³)	Br-82 (L), Na-24 (S)	D_p; V, %S	-	*Lelinski et al. (2002)*LELINSKI, D.; ALLEN, J.; REDDEN, L.; WEBER, A. Analysis of the residence time distribution in large flotation machines. Minerals Engineering, v. 15, p. 499-505, 2002.
Column (D_c=0.04 m, H_c=1.2 m)	Methylene blue (L)	J_g, Q_f	CSTR_s	Melo (2002)MELO, M. V. Tratamento de águas oleosas por flotação. 2002. Tese (Doutorado em Engenharia Química) - COPPE, Universidade Federal do Rio de Janeiro, Rio de Janeiro, 2002.
Tank (V=75 L)	Methylene blue (L)	J_g, J_f	CSTR_s	*Puget et al. (2004)*PUGET, F. P.; MELO, M. V.; MASSARANI, G. Modelling of the dispersed air flotation process applied to dairy wastewater treatment. Brazilian Journal of Chemical Engineering, v. 21, n. 2, p.229-237, 2004.
Industrial Mechanical (V=489 m³ and 508 m³)	KCl (L)	%S, V, Q_f	CSTR_s	*Lima et al. (2005)*LIMA, O. A.; LEAL FILHO, L. S.; SILVA, A. L.; MOURA, A. J. Distribuição de tempos de residência da polpa em células mecânicas de flotação. REM: Rev. Esc. Minas,Ouro Preto, v. 58, n. 3, p. 213-218, 2005.
Columns: pilot (D_c= 0.1 m, H_c=5.7 m) and industrial (V=196 m³)	MnCl₂, KBr (L) Irradiated ore (S)	J_g, Q_f, J_w, D_p	PFR_r, CSTR_s	Santos (2005)SANTOS, A. R. Estudo do comportamento dinâmico de colunas de flotação utilizando técnicas nucleares. 2005. 134 f. Dissertação (Mestrado em Ciência e Tecnologia das Radiações, Minerais e Materiais) - CDTN/CNEN, Belo Horizonte, 2005.
Columns and mechanical cells	-	-	CSTR_s, LSTS	Yianatos (2007)YIANATOS, J. Fluid flow and kinetic modelling in flotation related processes: columns and mechanically agitated cells - a review. Chemical Engineering Research and Design, v. 85, n. 12, p. 1591-1603, 2007.
Columns (D_c= 4 m, H_c=12 m)	Br-82 (L)	J_g,	CSTR, LSTS	*Massinaei et al. (2007)*MASSINAEI, M.; KOLAHDOOZAN, M.; NOAPARAST, M.; OLIAZADEH, M.; SAHAFIPOUR, M.; FINCH, J. A. Mixing characteristics of industrial columns in rougher circuit. Minerals Engineering, v. 20, p. 1360-1367, 2007.
Mechanical (V=130 m³)	CF3Br - Freon 13B1 (γ)	J_g	LSTS	*Yianatos et al. (2010)*YIANATOS, J.; CONTRERAS, F.; DÍAZ, F. Gas holdup and RTD measurement in an industrial flotation cell. Minerals Engineering, v. 23, p. 125 - 130, 2010.
Column (D_c= 0.1 m, H_c=1.68 m)	KCl (L)	J_g, J_f, C_f, %S	PFR	*Shukla et al. (2010)*SHUKLA, S. C.; KUNDU, G.; MUKHERJEE, D. Study of gas holdup and pressure characteristics in a column flotation cell using coal. Minerals Engineering, v. 23, n. 8, p. 636-642, 2010.
Mechanicals (V=160 m³)	Br-82 (L), Irradiated ore (S)	D_p, type of rotor	CSTR	*Yianatos et al. (2012)*YIANATOS, J.; BERGH, L.; PINO, C.; VINNETT, L.; MUÑOZ, C.; YAÑEZ, A. Industrial evaluation of a new flotation mechanism for large flotation. Minerals Engineering, v. 36-38, p. 262 - 271, 2012.
Mechanicals (V=100 - 300 m³)	Br-82 (L), Irradiated ore (S)	V, Q_f, D_p	CSTRs. LSTS	*Yianatos et al. (2015)*YIANATOS, J.; BERGH, C.; VINNETT, L.; PANIRE, I.; DÍAZ, F. Modelling of residence time distribution of liquid and solid in mechanical flotation cells. Minerals Engineering, v. 78, p. 69 - 73, 2015.
Column (D_c= 0.054 m, H_c=1.15 m)	KCl (L)	J_g, J_f, D_c	PFR_r	*Chegeni et al., 2015*CHEGENI, M. H.; ABDOLLAHY, M.; KHALESI, M. R. Column flotation cell design by drift flux and axial dispersion models. International Journal of Mineral Processing, v. 145, p. 83-86, 2015.
Columns V=2.5 - 180 m³)	NaCl, LiCl, Br-82 (L); Na-24, Sc-46 (S)	V, Q_f, D_c	CSTR_s PFR, LSTS	*Yianatos et al. (2017)*YIANATOS, J.; VINNETT, L.; PANIRE, I.; ALVAREZ-SILVA, M.; DÍAZ, F. Residence time distribution measurements and modelling in industrial flotation columns. Minerals Engineering, v.110, p. 139 - 144, 2017.

Variable	Conditions	t (s)	t_m (s)	σ_θ²	N
J_f (cm/s) - KCl	0.46	1200	1314.6	0.14	7.0
	0.62	900	908.0	0.09	11.1
	0.77	720	748.4	0.08	12.3
	0.93	600	670.1	0.08	12.2
J_f (cm/s) - MB	0.46	1200	1318.9	0.16	6.3
	0.62	900	989.4	0.11	9.3
	0.77	720	781.8	0.09	10.9
	0.93	600	693.1	0.09	11.2
C_f (mg/L)	0.0	900	908.0	0.09	11.1
	2.5	900	891.0	0.21	4.7
	5.0	900	848.0	0.18	5.5
	7.5	900	896.1	0.2	5.0
J_g (cm/s)	0.5	900	1032.1	0.09	10.7
	1.0	900	982.1	0.09	10.9
	1.5	900	908.0	0.09	10.6
	2.0	900	920	0.10	9.6

		Tanks-in-series model				Axial dispersion model
Variable	Condition	τ (s)	tm (s)	N	R²	t_m (s)	Pe	N_d*	R²
J_f (cm/s) KCl	0.46	1200	1258.49	7.56	0.9920	1312.6	12.86	0.078	0.9994
	0.62	900	893.81	9.59	0.9980	921.4	17.03	0.059	0.9996
	0.77	720	736.19	10.76	0.9980	757.9	19.67	0.051	0.9984
	0.93	600	650.00	11.85	0.9960	677.9	21.75	0.046	0.9992
J_f (cm/s) MB	0.46	1200	1271.44	8.84	0.9972	1320.0	15.73	0.064	0.9910
	0.62	900	964.24	11.39	0.9982	988.6	21.05	0.048	0.9974
	0.77	720	761.61	12.09	0.9990	779.4	22.46	0.045	0.9980
	0.93	600	681.80	13.23	0.9994	696.1	24.71	0.040	0.9970
C_f (mg/L)	0.0	900	893.81	9.59	0.9980	921.4	17.03	0.059	0.9996
	2.5	900	828.39	4.36	0.9752	898.0	6.2	0.161	0.9996
	5.0	900	791.07	4.63	0.9801	851.5	6.87	0.146	0.9998
	7.5	900	837.2	4.62	0.9789	901.3	6.82	0.147	0.9994
J_g (cm/s)	0.5	900	1001.08	12.90	0.9974	1023.1	24.01	0.042	0.9990
	1.0	900	950.4	11.19	0.9958	974.7	20.62	0.048	0.9996
	1.5	900	893.81	9.59	0.9980	921.4	17.03	0.059	0.9996
	2.0	900	885.0	8.86	0.9954	914.2	15.91	0.063	0.9996

E (t) = \frac{N^{N} \cdot t^{(N - 1)}}{t_{m}^{N} \cdot (N - 1)!} \cdot e^{- \frac{t \cdot N}{t_{m}}} = \frac{N^{N} \cdot t^{(N - 1)}}{t_{m}^{N} \cdot Γ (N)} \cdot e^{- \frac{t \cdot N}{t_{m}}}

[1] E-mails : 4 hcouto@cetem.gov.br, hudson.couto@ifrj.edu.br