ESTIMATION OF PARAMETERS AND STATES USING A BAYESIAN PARTICLE FILTER FOR THE SULFATE ION ADSORPTION PROCESS IN A FIXED BED COLUMN

Carvalho, Rhynara N. S. de; Guimarães, Damaris; Leão, Versiane A.; Dutra, Julio C. S.; Silva, Wellington B. da

doi:10.1590/0104-6632.20190363s20180059

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Process Systems Engineering • Braz. J. Chem. Eng. 36 (3) • Jul-Sep 2019 • https://doi.org/10.1590/0104-6632.20190363s20180059 copy

ESTIMATION OF PARAMETERS AND STATES USING A BAYESIAN PARTICLE FILTER FOR THE SULFATE ION ADSORPTION PROCESS IN A FIXED BED COLUMN

Authorship SCIMAGO INSTITUTIONS RANKINGS

Abstract

Ensuring that industrial effluents meet quality standards to be released into water bodies is still one of the major environmental concerns. Liquid phase adsoption in fixed bed collumns is one of the most known treatments. Measurements are needed to monitor the process. However, as they are often corrupted by noise from the measuring equipment, performing an accurate analysis becomes an important challenge. The present work demonstrates the effectiveness of particle filter Sampling Importance Resampling as a fast and robust tool for monitoring a problem of sulfate ion removal. Experimental measurements were used to validate the methodology and the particle filter (PF) performance was evaluated by means of error metrics, computational time and compared to the Unscented Kalman Filter. The results show that the PF provides sequentially very accurate estimates for the sulfate adsorption breakthrough curve.

Keywords:
State estimation; Particle filter; Modeling; Adsorption; Fixed bed

Parameter	Value	Unit
Cross sectional area of column (S)	0.000531	m²
Bed height (L)	0.09	m
Porosity of bed (ɛ)	0.35	-
Particle diameter (d_p)	0.00077	m
Particle radius (r_p)	0.000385	m
Density of liquid (ρ_L)	1·10³	kg/m³
Density of the bed (ρ_s)	1.134·10³	kg/m³
Viscosity of the mixture (µm)	53.49·10^-3	kg/(m·min)
Initial concentration (C₀)	170	mg/m³

Name	Model	Reference
External surface area per unit particle volume (a_v)	$a_{v} = \frac{3}{r_{p}}$	Ruthven (1984)
Coefficient of mass transfer in the external liquid film (K_e)	$k_{e} = \frac{S h \cdot D_{m}}{d_{p}}$	Muzic et al. (2010)
Schmidt Number	$S c = \frac{μ_{m}}{ρ_{L} D_{m}}$	Muzic et al. (2010)
Sherwood Number	$S h = 2 + 1, 1 R e^{0, 6} S c^{1 / 3}$	Ruthven (1984)
Reynolds Number	$R e = \frac{ρ_{L} v_{s} d_{p}}{μ_{m}}$	Muzic et al. (2010)

Description	Equations
Adsorption column model	$\frac{\partial C}{\partial τ} = \frac{1}{P e} \frac{\partial^{2} C}{\partial x^{2}} - \frac{\partial C}{\partial x} - ρ_{L} \frac{1}{ε} \frac{\partial q}{\partial τ}$
Initial condition	$C (0, x) = 0$
Boundary conditions	$\frac{\partial C}{\partial x} = P e [C (τ, 0) - C_{0}]$ at $x = 0$
Boundary conditions	$\frac{\partial C}{\partial x} = 0$ at $x = 1$

Name	Model	Reference
Surface velocity (v_s)	$v_{s} = \frac{Q}{S}$	Ruthven (1984)
Interstitial velocity (u₀)	$u_{0} = \frac{v_{s}}{ε}$	Ruthven (1984)
Axial dispersion (D_L)	$\frac{D_{L}}{u_{0} d_{p}} = \frac{20}{ε} (\frac{D_{m}}{u_{0} d_{p}}) + \frac{1}{2}$	Ruthven (1984)
Molecular diffusivity (Dm)¹	$D_{S O_{4}^{-}} = \frac{(Z_{+} + \|Z_{-}\|) D_{N a^{+}}^{0} D_{S O_{4}^{-}}^{0}}{Z_{+} D_{N a^{+}}^{0} + \|Z_{-}\| D_{S O_{4}^{-}}^{0}}$	Muzic et al. (2010).

N	Position in the column (x)	Dimensionless time (τ)
N	Position in the column (x)	0.7174	1.5245	2.3316
10	0.5	6.269029	0.676022	0.07014
50		1.245502	0.133156	0.01002
100		0.595073	0.061457	0
1,000		0	0	0
10	1.0	8.431953	1.929116	0.161714
50		0.332840	0.044863	0
100		0.073964	0.010107	0
1,000		0	0	0

N_p	σ_m	CPU time (s)	RMS	MAD
10	5%	0.4837	0.3703	0.2350
	10%		0.3107	0.1958
	30%		0.1267	0.0729
100	5%	2.3458	0.3386	0.2108
	10%		0.2646	0.1550
	30%		0.0118	0.0078
500	5%	11.0629	0.3164	0.1985
	10%		0.1737	0.0940
	30%		0.0034	0.0028
1000	5%	18.7912	0.3065	0.1912
	10%		0.1857	0.1041
	30%		0.0044	0.0033

Filter	σ_m	CPU time (s)	Error metrics
Filter	σ_m	CPU time (s)	RMS	MAD
PF	5%	11.06	0.3164	0.1985
	10%		0.1737	0.0940
	30%		0.0034	0.0028
UKF	5%	1.49	0.2098	0.1410
	10%		0.0850	0.0558
	30%		0.0229	0.0164

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[1] * Corresponding author: Wellington B. da Silva - E-mail: wellingtonufes@gmail.com