7842
Ind. Eng. Chem. Res. 2009, 48, 7842–7846
Approximate Method for Designing a Primary Settling Tank for Wastewater Treatment Gloria Martı´nez-Gonza´lez, Herbert Lorı´a-Molina, David Taboada-Lo´pez, Francisco Ramı´rez-Rodrı´guez, Jose´ Luı´s Navarrete-Bolan˜os, and Hugo Jime´nez-Islas* Departamento de Ingenierı´a Quı´mica-Bioquı´mica, Instituto Tecnolo´gico de Celaya, AVe. Tecnolo´gico y Garcı´a Cubas s/n, Celaya, Gto. 38010, Me´xico
An approximate method based on an analogous Fick’s law mass transport model was derived to predict the concentration profiles of suspended particles in a primary sedimentation tank for wastewater treatment as a function of time and column height. A pilot sedimentation column was constructed to test wastewater sedimentation and obtain the necessary data for the proposed model. The examined variables included the total suspended solids concentration, the sedimentation height, and the time elapsed. Computer code was developed to estimate the dispersion coefficient present in the sedimentation model from experimental data via least-squares regression, resulting in a relative error of 3.705%. The proposed model was validated by good agreement with reported data in the literature. The basic dimensions for designing a primary sedimentation tank were obtained based on the experimental data from wastewater samples obtained by a wastewater collector located in the industrial zone of Celaya City, Me´xico. This methodology can be successfully applied for designing primary sedimentation tanks for wastewater treatment facilities. 1. Introduction Sedimentation is a crucial stage of domestic and industrial wastewater treatment in which suspended particles are separated from a liquid by gravity. Consequently, sedimentation is one of the most frequently used unit operations for the clarification of residual water. The physical phenomenon associated with the gravitational precipitation of solid particles in a liquid has been widely studied. The equation describing the velocity of sedimentation, formulated by Stokes in 1851,1 is the starting point for analyzing sedimentation. From this incipient formulation, diverse assays have been performed to find a mathematical model to explain sedimentation. Hazen2 introduced the surface loading concept in 1904 and analyzed sedimentation factors for solid particles contained in diluted water solutions. In 1952, Kynch3 proposed a kinetic theory of sedimentation based on concentration changes in a suspension. In recent years, studies have been conducted examining the properties of sedimentation processes for ideal and flocculating suspensions at Universidad de Concepcio´n in Chile and Universita¨t Stuttgart in Germany.4 In spite of these advances that are proposed to unify studies on sedimentation of dispersed and flocculating suspensions, there are not suitable equations to simulate the settling phenomenon in wastewater archetypes and facilitate the design of sedimentation tanks.4 Therefore, exhaustive experimental measurements are required for the construction of solid-removal curves for the calculation of dimensions for primary settlings.5 Additionally, several experiments are required to obtain solid-removal contour plots at different heights and times, which are also required for construct charts that describe the total solid-removal percentage in the tank at a given time. This procedure is laborious because it is necessary to have a large amount of experimental data to obtain plots with acceptable accuracy.6 Rigorous models based in computational fluid dynamics (CFD) are used to predict flow patterns and suspended solid distributions within sedimentation tanks.7,8 These are normally used to find the relationship between the tank hydraulics and * To whom correspondence should be addressed. Phone: +52 (461) 6117575. Fax: +52 (461) 6117979. E-mail:
[email protected].
the process efficiency. The use of CFD-based models has not been common due to the inherent complexity of the corrected Navier-Stokes equations for turbulent flow8,9 and the costs associated with the specialized hardware and software required; however, there have been reports of simplified models including empirical parameters.10-12 Currently, few industrial organizations use CFD techniques to study flow phenomena in their wastewater treatment facilities due to the high cost of commercial licenses of CFD software.13 Thus, the objective of this study was to develop an approximate and feasible mathematical model that allows for the prediction of concentration profiles of suspended particles based on time and height measurements in a sedimentation column, without the need for conventional experimental-plotting procedures. Model Description. Wastewater is considered as an incompressible and Newtonian fluid within the sedimentation column shown in Figure 1, containing particles of equal size and density (component A). These particles flow toward the bottom of the column by gravity. The settling velocity is assumed to be constant, and flocculation effects are assumed to be negligible. Equation 1 describes a microscopic mass balance using Fick’s law adapted to macrodispersion, as reported by Johnson and DePaolo,14 Beg et al.,15 and Ginn et al.16 The dispersion coefficient, DE, includes turbulence effects caused by particles and gravity and was considered constant. Thus, the continuity equation for component (A) was as follows. ∂CA + (v · ∇CA) ) DE∇2CA + RA ∂t
(1)
where v ) Vzk. Additional assumptions for the proposed sedimentation model are as follows: 1. Solid dispersion is equal for particles of equal size throughout the liquid. 2. Solid dispersion occurs in the vertical direction, z; thus, the particle concentration is only a function of the time and column height, CA ) CA(z, t).
10.1021/ie801869b CCC: $40.75 2009 American Chemical Society Published on Web 07/24/2009
Ind. Eng. Chem. Res., Vol. 48, No. 16, 2009
| |
dVzFL ) 0.873699 < 2 µL
Re )
7843
(7)
2. Experimental Section
Figure 1. Schematic of analyzed system, including initial and boundary conditions in the sedimentation column.
3. There are no chemical reactions, RA ) 0; at t ) 0, CA ) CA0; at z ) L and t > 0, CA ) 0. At z ) 0, the solid flux is equal to zero because the bottom of the column is closed. 4. Radial dispersion is negligible due to a particle density greater than 1 and a height/diameter ratio greater than 10. On the basis of these assumptions, eq 1 reduces to eq 2. ∂CA ∂2CA ∂CA ) DE + Vz 2 ∂t ∂z ∂z
(2)
The term Vz is referred to as the terminal velocity of particles and takes into account Stokes’s law; for its calculation, it was necessary to define values for FS, FL, d, and µL. FS was defined as 2.65 kg/m3, a typical density of sand;5 FL and µL at 20 °C were defined as 1000 kg/m3 and 0.001014 kg/m · s, respectively; and, the particle diameter, d, was defined as 0.0001 m (100 µm), an average value based on reported sedimentation processes.6,19 Here, Vz was obtained from Stokes’s law: g(FS - FL)d2 Vz ) ) -0.008859 m/s 18µ
(3)
In Equation 2, the velocity, Vz, is negative due to effects from the model solution, which is also considered by Bustos et al.1 in their proposed dynamic models. Equation 2 has been obtained by Pritchard17 and Mucha et al.18 using their own reference system, which is used in this work. Initial conditions: t ) 0,
CA(z, 0) ) CA0, 0 < z < L
(4)
Boundary condition 1: z ) 0,
ζ)
z L
t tC
τ)
C* )
CA CA0
(8)
Substituting these groups results in a dimensionless form of eq 2: ∂2C* ∂C* ∂C* )R 2 +β ∂τ ∂ζ ∂ζ
(9)
where the dimensionless coefficients are given by R)
β)
DEtC
(10)
L2 VztC L
(11)
and the boundary conditions become ∂C* ∂ζ
ζ ) 0,
|
ζ)0
) 0,
τ>0
(12)
Using Kynch’s3 boundary condition at ζ ) 1, we have ∂CA ∂z
|
z)0
) 0,
t>0
(5)
ζ ) 1,
τ ) 0, CA(L, t) ) 0,
C*| ζ)1 ) 0,
τ>0
(13)
0
