Environ. Sci. Technol. 2008, 42, 1718–1723
Drag Coefficient of Porous and Permeable Microbial Granules YANG MU, TING-TING REN, AND HAN-QING YU* Department of Chemistry, University of Science & Technology of China, Hefei, 230026, China, and Advanced Water Management Centre, The University of Queensland, St. Lucia, QLD, 4072, Australia
Received October 26, 2007. Revised manuscript received December 14, 2007. Accepted December 17, 2007.
Settling velocity of microbial aggregates, such as anaerobic and aerobic granules, in biological wastewater treatment systems is highly related to their drag coefficient. In this work a new approach, taking the porosity and the permeability into account, was established to evaluate the drag coefficient of porous and permeable microbial granules. The effectiveness of this approach was demonstrated by the experimental results with both the anaerobic and the aerobic granules. The drag coefficient of the microbial granules was found to be less than that of smooth rigid spheres and Biofilm-covered particles. In addition, this study demonstrates that the drag coefficient of microbial granules depended heavily on their permeability and porosity. A fractal-cluster model was found to be able to predict the distribution of the primary particles in the microbial granules.
defined particle shape (12, 13). In comparison to these correlations for specific particle shapes, very few studies appeared for the irregularly shaped particles, especially the porous and permeable aggregates (14, 15). This is due to the extreme difficulty of taking the permeability into account when the drag coefficient is calculated. On the other hand, in most of the models concerning permeability, the homogeneous distribution of pores is assumed within the aggregates. Unfortunately, the experimental results demonstrate that the models are not able to accurately predict the aggregate settling behavior when this assumption is adopted (16). Therefore, the primary objective of this study was to provide a reliable approach to evaluate the drag coefficient of the porous and permeable microbial granules. A new approach was realized by taking the porosity and permeability of the granules into consideration. Both the anaerobic and the aerobic granules were adopted to demonstrate the effectiveness of this new approach. The results from this study may provide valuable information for the operation and the design of a granule-based bioreactor for wastewater treatment. Theoretical Approach. The force balance for a granule steady in an infinite medium can be described as follows (17): Fg ) Fb + Fd
where Fg is gravity, Fb is buoyant, and Fd is drag force of a granule. For a porous and permeable microbial granule, Fg, Fb, and Fd can be respectively expressed as follows: Fg ) VaFlg + (1 - )VaFpg
(2)
Fb ) FlVag
(3)
Introduction Since the concept of the upflow anaerobic sludge blanket (UASB) reactor was developed in the late 1970s, it has become the most popular high-rate reactor for anaerobic treatment of wastewater throughout the world (1). Settling velocity of methanogenic granules in the UASB reactor is a critical factor that regulates the solid–liquid separation and effluent quality (1). Recently, aerobic granules have been extensively investigated and applied for treating wastewaters from various industries such as malting (2), dairy (3), and soybeanprocessing (4), as well as municipal wastewater (5). Both anaerobic and aerobic granules are found to be porous and permeable (6, 7). The settling velocities of microbial granules are important for tracking the transport and the removal of particles in natural waters and in engineered systems such as the wastewater treatment plant clarifiers. Since the settling velocity of a particle depends greatly on its drag coefficient, a reliable evaluation of the drag coefficient of porous and permeable microbial granules is highly desirable (8, 9). In earlier investigations, the drag force experienced by spheres moving through a fluid was the main focus. Extensive sets of data were collected and combined with theoretical work to formulate several empirical correlations for the drag coefficient (10, 11). Several studies regarding the drag coefficient of irregular particles mainly addressed a limited number of solid shapes and, in most cases, formulated the particle free-falling velocity and drag with respect to a well * Corresponding author. Fax: +86 551 3601592; E-mail:
[email protected]. 1718
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(1)
U2
Fd )
Fl AΩCd 2
(4)
where is the porosity, Va is the encased volume, Fl is the density of a liquid, Fp is the density of a granule, g is the gravitational constant, U is the settling velocity, Cd is the drag coefficient of a particle in a liquid, A is the projected area of an object, and Ω is the reduction of the drag force. Substituting eqs 2–4 into eq 1 gives VaFlg + (1 - )VaFpg ) FlVag +
FlU 2ΩACd 2
(5)
In eq 5, A ) π / 4d2 and Va ) π / 6d3 . Thus, eq 5 can be rewritten into π FlU 2 d 2ΩCd π 4 ) (1 - ) d 3(Fp - F1)g 2 6 Cd )
4 × (1 - ) × d × (Fp - Fl) × g 3 × Fl × Ω × U2
(6) (7)
As shown in eq 7, the porosity and the permeability of a granule must be taken into account to calculate its drag coefficient. Porosity of Microbial Granules. For a microbial granule, it is assumed that there are N identical cells. Each cell has a dry solid mass of md when the water inside the cell is excluded and a wet mass of mc with density of Fp when the interior liquid is included. From the encased volume of the 10.1021/es702708p CCC: $40.75
2008 American Chemical Society
Published on Web 02/01/2008
TABLE 1. Three Most Common Permeability Modelsa
a
model
permeability function
equation
Brinkman model Carman-Kozeny model Happel model
κB ) d2g / 72 (3 + 4 / 1 - - 3√ 8 / 1- - 3) κC ) d2g / 180 ( 2 / (1 - )2 ) κH ) d2g / 18 ( 3 - 4.5γ + 4.5γ5 - 3γ6 / γ3(3 + 2γ5) )
16a 16b 16c
Note: γ ) (1 - )1⁄ 3 .
granule, Va ) πd3/6, and the total cell volume within the granule, Vc ) Nmc/Fp, the granule has the following porosity: )1-
Vc 6Nmc )1Va πF d 3
(8)
p
The total cell mass of the granule in water, Nmc, can be calculated from Nmc ) Nfmd ) fWd, where f is a dimensionless ratio between the wet mass and dry mass of the cells. Therefore, eq 8 can be written as follows: 6f Wd
(9)
FIGURE 1. Images of the anaerobic (a) and the aerobic (b) granules.
Aggregates, including microbial granules, are fractal and can be characterized by a fractal dimension (18) as follows:
application of eq 14, nevertheless, requires the value of granule permeability, κ, whose evaluation is usually based on the permeability models. A number of formulas have been utilized for calculating the permeability of a granule, κ. The most commonly used correlations for granules, summarized in Table 1, are the Carman-Kozeny equation (19, 21), the Brinkman equation (19, 22), and the Happel equation (23). All the equations are in the form of κ ) d2g×f(), where f() is a function of granule porosity and dg is the size of the fractal generator of the granules.
)1-
πFpd 3
Wd ) BdD
(10)
where D is the fractal dimension. For a Euclidean structure, D ) 3, while for fractal granules, D < 3. By substituting eq 10 into eq 9, the porosity of a granule can be expressed with the following equation: )1-
6fB D-3 d πFp
(11)
Materials and Methods
Permeability of Microbial Granules. The porous fractal structure of a granule may permit a significant intra-aggregate flow, resulting in a reduced drag force and thus an increased settling velocity as compared with that of identical but impermeable sphere predicted by Stokes’ law (19). The reduction in the drag force (Ω) is defined as below (20): Ω)
resistance experienced by a permeable sphere resistance experienced by a permeable sphere (12)
The Ω takes into account of the advection flow through the aggregate’s interior, which is unity for an impermeable sphere. An aggregate with a high permeability would experience a strong advection flow through its interior, whereas Ω could be far less than unity. Neale et al. (20) analyzed the problem of a swarm of permeable flocs moving with the same velocity in an infinite medium. Based on a cell model, they concluded that under the condition of creeping flow, the parameter Ω and the permeability of a floc, κ, could be related by the following equation: Ω)
2ξ3 - ξ2tanhξ 2ξ3 + 2ξ - 3tanhξ
(13)
where the dimensionless permeability of the aggregate, ξ, is calculated from the following equation: ξ)
d 2√k
(14)
From eq 13, it can be seen that the parameter Ω is solely a function of the permeability factor, ξ. For impermeable granules, as ξ approaches infinity, Ω ) 1. As the granule permeability increases, both ξ and Ω also decrease. A direct
Microbial Granules. As shown in Figure 1a, the anaerobic granules used in this study were collected from a laboratoryscale UASB reactor treating the sucrose-rich wastewater at 10 g-COD/L (COD ) chemical oxygen demand), hydraulic retention time of 12 h, temperature of 37 ( 1 °C, and pH of 7.0 ( 0.1. The aerobic granules shown in Figure 1b were collected from a sequencing batch reactor treating the sucrose-rich wastewater at 1.5 g-COD/L, hydraulic retention time of 6 h, temperature of 25 ( 1 °C, and pH of 7.2 ( 0.5. Analysis. The density (Fp) of both the aerobic and the anaerobic granules was determined according to the method described by Zheng et al. (24). The fractions of granules with different diameters were separated using sieves. The granule size was measured using an image analysis system (Imagepro Express 4.0, Media Cybernetics Inc., U.S.) with an Olympus CX41 microscope and a digital camera (C5050, Olympus Co., Japan). The granules were dried in an oven with 101 °C for 2 h and the dry mass (Wd) of the granules was weighed on an analytical balance (AD2B, Perkin-Elmer Inc., U.S.). The ratio (f) of the wet mass to the dry mass of the granules was estimated as 3.45 according to Li and Yuan (25). The settling velocity of the granules was measured using a glass-column, which was 40 cm in height to ensure that the terminal settling velocity could be reached, and 4.0 cm in internal diameter to minimize the wall effect on granule settling (26). In addition, the horizontal distance between the granule and the column was kept at the same value in the settling tests.
Results Porosity of Granules. The relationship between the dry mass and the granule diameter is illustrated in Figure 2a. The diameter (d) of the anaerobic and the aerobic granules varied VOL. 42, NO. 5, 2008 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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FIGURE 2. Dry mass (a) and porosity (b) as a function of the size for the anaerobic and the aerobic granules. from 0.10 to 0.50 cm and from 0.11 to 0.38 cm, respectively. Their dry mass (Wd) ranged from 0.02 to 6.90 mg per granule and from 0.02 to 0.53 mg per granule, respectively. Based on the slope and the intercept of the line in Figure 2a, the fractal dimension D of the anaerobic and aerobic granules was calculated to be 2.83 ( 0.10 and 2.65 ( 0.20, and the constant B to be 0.05 and 0.01, respectively. Thus, the porosity of the anaerobic (AN) and the aerobic (A) granules could be calculated from the following equation: )1 -
)1 -
6 × 3.45 × 0.05 2.86-3 6fB D-3 d )1) 1 - 0.31 × d -0.14 d πFp 3.14 × 1.06 (15a)
6fB D-3 6 × 3.45 × 0.01 2.65-3 d )1) 1 - 0.064 × d -0.35 d πFp 3.14 × 1.02 (15b)
Figure 2b shows the relationship between the porosity and the size of the two granules. Their porosity increased with the increasing granular size, within a range of 0.57–0.66 and 0.85–0.92, respectively. Permeability of Granules. Table 1 lists the three most common models for calculating the hydraulic permeability of a granule. The Carman-Kozeny permeability model is a semiempirical equation. Its disadvantage is that it is only valid when the porosity is less than 0.5 (23). Thus, it cannot be used to predict the permeability for high-porosity granules. Therefore, in our study the Brinkman model (eq 16a) and Happel model (eq 16c) were adopted to estimate the permeability of the anaerobic and aerobic granules. The hydraulic permeability, κ, and the permeability factor, ξ, for the granules can be derived by substituting eq 15a for in
the permeability equations given in Table 1, as summarized in Table 2. Two parameters (d and dg) in eqs 17a, 17c and 18a, 18c indicate that the permeability of both anaerobic and aerobic granules was related to the size of the granule itself and its fractal generator. Two approaches have emerged to model the distribution of the primary particle in granules: the single-particle-fractal model and the cluster-fractal model (25). In the singleparticle-fractal model a uniform distribution of the small spheres within the granule is assumed (16). This means that the size of the fractal generator is equal to the diameter of the primary particles in the granule, i.e. dg ) dp. Li and Ganczarzyk (17) selected dp to be between 1 and 10 µm. The most important feature of a fractal granule in the clusterfractal model is that the primary particles within the granule are not uniformly distributed within the volume of the granule, but are instead separated into smaller principal clusters that are themselves fractal objects formed through particle coagulation. These principal clusters are considered as the fractal generators of granule, i.e., dg ) dc, where dc is the size of the principal clusters, around 1000 µm (18). In this study various dg values were selected to compare these two models, e.g., dg values of 1, 5, and 10 µm for the single-particle-fractal model while dg value of 1000 µm for the cluster-fractal model. The hydraulic permeability, κ, and the permeability factor, ξ, calculated from eqs 17a, 17c and 18a, 18c as a function of the granule size are, respectively, shown in Figures 3 and 4. Both the hydraulic permeability and the permeability factor increased slightly with the increasing granule size. In addition, the hydraulic permeability increased as the size of the fractal generators increased (Figure 3), whereas the permeability factor had an opposite changing trend (Figure 4). The reductions of the drag force, Ω, estimated from eq 13, as a function of the granule size, are shown in Figure 5. The Ω values of both anaerobic and the aerobic granules were near to unity for the single-particle-fractal model at various granular sizes, implying that the granules were impermeable. Since previous experiments have demonstrated that both anaerobic and aerobic granules were permeable (4, 6), the single-particle-fractal model was not appropriate for estimating the Ω values. However, their Ω value for the cluster-fractal model was both less than unity, and increased with an increase in granule size, as shown in Figure 5. Drag Coefficient of Granules. As shown in Figure 6, the terminal settling velocity of the anaerobic and the aerobic granules increased with an increase in granule size, from 0.4 to 6.6 cm/s and 0.6 to 1.6 cm/s, respectively. The Reynolds number of the granules could be calculated from eq 19: Re )
FldU µ
(19)
The Reynolds number of the anaerobic and the aerobic granules, also shown as a function of the granular size in Figure 6, increased as the granular size increased.
TABLE 2. Permeability Function for the Anaerobic and Aerobic Granules granule
model
hydraulic permeability
0.14 2 anaerobic Brinkman κ -0.12-0.59)2 B-AN ) 0.18×d g×(√d Happel κH-AN ) 0.18×d 2g×( d 0.37 - d0.32 + 0.21×d0.14 - 0.1×d 0.09 / d 0.23 + 0.09 )
aerobic
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dimensionless permeability factor ςB-AN ) 1.15×d dg×(√d0.14-0.12-0.59) ςN-AN ) 1.2×d / dg×( d 0.37 - d0.32 + 0.21×d 0.14 - 0.1×d 0.09 / d 0.23 + 0.09 )1⁄2
/
0.35 2 Brinkman κ -0.024-0.27)2 ξB-A ) 0.54×d d g×(√d 0.35 - 0.02 - 0.27) B-A ) 0.87×d g×(√d 0.82 0.35 2 ( 0.93 Happel - 0.6×d + 0.015×d - ςH-A ) 0.54×d / d g×( d 0.93 - 0.6×d 0.82 + κH-A ) 0.87×d g× d 0.004×d 0.23 / d 0.58 + 0.007 ) 0.015 × d 0.35 - 0.001×d 0.12 / d 0.58 + 5.83 )1⁄2
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equation 17a 17c
18a 18c
FIGURE 3. Hydraulic permeability as a function of the size for the anaerobic (a) and the aerobic (b) granules (the thin lines: Brinkman model; the thick lines: Happel model). After their Ω value and terminal settling velocity (U) were determined, the drag coefficient of the anaerobic (Cd-AN) and the aerobic (Cd-A) granules could be calculated from eqs 7, 13, and 15a: Cd-AN )
4 × 0.31 × d -0.14 × d × (1.06 - 1) × 980 3 × 1.0 × Ω × U2 ) 24.3 ×
Cd-A )
d 0.86 (20a) Ω × U2
4 × 0.064 × d -0.35 × d × (1.02 - 1) × 980 d 0.65 ) 1.7 × 2 3 × 1.0 × Ω × U Ω × U2 (20b)
As shown in Figure 7, the drag coefficient of both anaerobic and the aerobic granules decreased with an increase in the Reynolds number. An empirical correlation has been used to estimate the drag coefficients from the Reynolds number (27):
(
Cd ) a +
b Re 0.5
)
2
(21)
In this study, two parameters in eq 21, a and b, were estimated by using nonlinear regression as 0.24 and 5.6 for the anaerobic granules, 0.38 and 1.68 for the aerobic granules. Thus, eq 21 could be expressed as follows:
( (
Cd-AN ) 0.24 + Cd-A ) 0.38 +
) )
5.6 Re 0.5
1.7 Re 0.5
2
2
(22a) (22b)
FIGURE 4. Dimensionless permeability factor as a function of the size for the anaerobic (a) and the aerobic (b) granules (the thin lines: Brinkman model; the thick lines: Happel model). The high regression coefficient suggests that the two empirical correlations were able to appropriately describe the relationship between the drag coefficients and the Reynolds number of the anaerobic and the aerobic granules.
Discussion In the present study, an approach was established to evaluate the drag coefficient of porous and permeable granules. The established model was able to describe the drag characteristics of both the anaerobic and the aerobic granules. In the previous studies, the porosity and the permeability of microbial granules have never been considered to estimate their drag coefficient (9, 28). However, this study demonstrates that the drag coefficient of microbial granules depends heavily on their permeability and porosity. Three different permeability correlations have typically been used to model the permeability of aggregates (19, 21, 23). The Carman-Kozeny equation was found not appropriate to estimate the permeability of both the anaerobic and the aerobic granules. This result agrees with the simulation results of Lee et al. (19). They compared the Brinkman, Happel, and Carman-Kozeny permeability equations using computer simulations. From the simulation they recommended that the Brinkman or Happel equation is the most appropriate for aggregates in terms of a better convergence. The differences in the permeabilities predicted using the Brinkman and the Happel equations are minor. The Brinkman equation predicts a slightly higher permeability than the Happel equation for the two types of granules. Considering such a minor difference, either the Brinkman equation or the Happel equation can be used to calculate their permeability. The comparison between the simulated results and the experimental data demonstrates that the fractal-cluster model could predict the distribution of the primary particles in the VOL. 42, NO. 5, 2008 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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FIGURE 7. Drag coefficients as a function of Reynolds number for the anaerobic (a) and the aerobic (b) granules. FIGURE 5. Ω as a function of the size for the anaerobic (a) and the aerobic (b) granules (the thin lines: Brinkman model; the thick lines: Happel model).
therefore must be composed of smaller fractal clusters that similarly obey a fractal scaling relationship. The successful application of the cluster-fractal model is attributed to the large pores created by the clusters. These large pores make the granules highly permeable, even though there is relatively little flow within the smaller clusters. The following empirical equation has been proposed to calculate Cd for smooth, rigid, and spherical particles (27).
(
Cd ) 0.63 +
4.8 Re 0.5
)
2
(23)
For biofilm-covered particles, the following empirical correlations have been suggested (9, 29):
(
Cd ) 0.955 +
(
Cd ) 0.8 +
FIGURE 6. Settling velocity and the Reynolds number as a function of the size for the anaerobic (a) and the aerobic (b) granules. microbial granules, i.e., not a uniform distribution of the primary particles in the granules. The microbial granules are formed through the coagulation of smaller aggregates, and 1722
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7.28 Re 0.5
6.1 Re 0.5
)
) 2
2
(24a) (24b)
Figure 8 compares the drag coefficient values calculated using eqs 22a and –24a for smooth-rigid-spherical particles, biofilmcovered particles, anaerobic, and aerobic granules. The drag coefficient of the biofilm-covered particles is clearly higher than that of the smooth rigid spheres over the whole range of Re investigated. The biofilm-covered particles have rough surface, which adds to the drag (9). However, the drag coefficient of the anaerobic and the aerobic granules is less than that of the smooth rigid spheres, as shown in Figure 8. This is mainly attributed to the fact that liquid can flow through the porous and permeable microbial granules, which reduces their drag. The lower fractal dimension of the anaerobic granules than that of the aerobic suggests that the anaerobic granules are more regular and compact. This led to their lower porosity (Figure 2) and permeability (Figure 5). As a result, the drag
FIGURE 8. Comparison of drag coefficient values for smooth-rigidspherical particles, biofilm-covered particles, anaerobic, and aerobic granules. coefficient of the anaerobic granules was higher than that of the aerobic ones with the same Re, as shown in Figure 8.
Acknowledgments We thank the Natural Science Foundation of China (20577048, 50625825, and 50738006), and the National Basic Research Program of China (2004CB719602) for the partial support of this study.
Appendix A Nomenclature Fg Fb Fd d dp dc dg F g mc md A B Va Vc Cd Cd-AN Cd-A D N Re U Wd Fp Fl AN A µ Ω ζ κ R2
gravity of granule ((g cm)/s2) buoyant of granule ((g cm)/s2) drag force of granule ((g cm)/s2) diameter of a granule (cm) size of the primary particles (cm) size of the principal clusters (cm) size of the fractal generator (cm) ratio of the wet mass to the dry mass of bacterial cells gravitational constant (cm/s2) cell mass of a bacterium (g) dry mass of a bacterium (g) projected area of an object (cm2) constant encased volume of a granule (cm3) volume of total cells within a granule (cm3) drag coefficient of a particle in a liquid drag coefficient of anaerobic granules drag coefficient of aerobic granules fractal dimension number of bacterial cells in a granule Reynolds number settling velocity (cm/s) dry mass of a granule (g) density of a granule (g/cm3) density of a liquid (g/cm3) porosity porosity of anaerobic granules porosity of aerobic granules fluid viscosity (g/cm/s) reduction of the drag force dimensionless permeability factor Permeability (cm2) regression coefficient
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