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Mathematical modeling and analysis of the flocculation process in chambers in series

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In this study, the flocculation process in continuous systems with chambers in series was analyzed using the classical kinetic model of aggregation and break-up proposed by Argaman and Kaufman, which incorporates two main parameters: K a and K b. Typical values for these parameters were used, i. e., K a = 3.68 × 10−5–1.83 × 10−4 and K b = 1.83 × 10−7–2.30 × 10−7 s−1. The analysis consisted of performing simulations of system behavior under different operating conditions, including variations in the number of chambers used and the utilization of fixed or scaled velocity gradients in the units. The response variable analyzed in all simulations was the total retention time necessary to achieve a given flocculation efficiency, which was determined by means of conventional solution methods of nonlinear algebraic equations, corresponding to the material balances on the system. Values for the number of chambers ranging from 1 to 5, velocity gradients of 20–60 s−1 and flocculation efficiencies of 50–90 % were adopted.

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ε :

Total energy dissipated per unit mass of fluid (L 2 T −2)

E :

Flocculation efficiency (−)

G :

Average velocity gradient (T −1)

K a :

Aggregation coefficient (−)

K b :

Break-up coefficient (T −1)


Coefficient related to floc resistance (−)

m :

Number of chambers in series

n :

Number of primary particles per unit volume at time t for batch system or at the inlet stream for continuous system (L −3)

n 0 and n m :

Number of primary or destabilized particles in the water before flocculation and in the effluent of the mth chamber, respectively (L −3)

n 0 :

Number of primary particles at time t = 0 (L −3)

n i and n i−1 :

Concentrations of primary particles in the output of the ith and (i − 1)th flocculation chambers, respectively (L −3)

Q :

Flow of water (L 3 T −1)

R :

Flocculation efficiency expressed by the ratio n 0 /n m (−)

T :

Total hydraulic retention time in the set of m chambers in series (T)

V :

Volume of the flocculation chamber (equal for all chambers) (L 3)

ν :

Kinematic viscosity of the fluid (L 2 T −1)


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The authors are particularly grateful to PROPe, Unesp (Provost of Research of State University of Sao Paulo, UNESP) for the payment of the translation and edition costs by means of the PROINTER program (inf. 028/2012).

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Correspondence to Rodrigo Braga Moruzzi.

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Moruzzi, R.B., de Oliveira, S.C. Mathematical modeling and analysis of the flocculation process in chambers in series. Bioprocess Biosyst Eng 36, 357–363 (2013). https://doi.org/10.1007/s00449-012-0791-4

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  • Mathematical modeling
  • Kinetics
  • Flocculation
  • Water and effluent treatment