Environ. Sci. Technol. 2010, 44, 8989–8995
Computational Fluid Dynamics Investigation of Turbulence Models for Non-Newtonian Fluid Flow in Anaerobic Digesters BINXIN WU Philadelphia Mixing Solutions Ltd., Palmyra, Pennsylvania 17078, United States
Received March 31, 2010. Revised manuscript received October 17, 2010. Accepted October 18, 2010.
In this paper, 12 turbulence models for single-phase nonNewtonian fluid flow in a pipe are evaluated by comparing the frictional pressure drops obtained from computational fluid dynamics (CFD) with those from three friction factor correlations. The turbulence models studied are (1) three high-Reynolds-number k-ε models, (2) six low-Reynolds-number k-ε models, (3) two k-ω models, and (4) the Reynolds stress model. The simulation results indicate that the Chang-Hsieh-Chen version of the low-Reynolds-number k-ε model performs better than the other models in predicting the frictional pressure drops while the standard k-ω model has an acceptable accuracy and a low computing cost. In the model applications, CFD simulation of mixing in a full-scale anaerobic digester with pumped circulation is performed to propose an improvement in the effective mixing standards recommended by the U.S. EPA based on the effect of rheology on the flow fields. Characterization of the velocity gradient is conducted to quantify the growth or breakage of an assumed floc size. Placement of two discharge nozzles in the digester is analyzed to show that spacing two nozzles 180° apart with each one discharging at an angle of 45° off the wall is the most efficient. Moreover, the similarity rules of geometry and mixing energy are checked for scaling up the digester.
1. Introduction Anaerobic digestion is a biological process that treats biodegradable materials such as agricultural manure and slurry, food waste, and sewage sludge to produce a methanerich biogas. It has long been recognized that effective mixing is a critical physical operation in achieving optimum process performance for anaerobic digestion. Under most circumstances the mixing flow in anaerobic digesters is turbulent, and the accuracy of a flow prediction is dependent on the turbulence model being used because it has a pronounced effect on the flow fields. Also, a non-Newtonian fluid such as municipal wastewater sludge or liquid manure may exhibit a flow behavior completely different from that of a Newtonian fluid. Although numerous turbulence models are available in the literature, there are no quantitative guidelines for choosing appropriate turbulence models that characterize non-Newtonian fluid flow in anaerobic digesters. The performance of a turbulence model can be evaluated by comparing the frictional pressure drop (or friction factor) obtained from computational fluid dynamics (CFD) with that * Author to whom correspondence should be sent: E-mail:
[email protected]; phone: 717-832-8857; fax: 717-832-1740. 10.1021/es1010016
2010 American Chemical Society
Published on Web 11/03/2010
from a correlation analysis or an experimental measurement. Metzner and Reed (1) introduced the concept of a generalized Reynolds number for power-law fluids that relates to the consistency coefficient and the power-law index. Subsequently, Dodge and Metzner (2) proposed an implicit equation for predicting the friction factor of viscous fluids in turbulent pipe flow. Later, several researchers (3-5) developed explicit correlations for calculating the friction factor. These correlations have been widely used in the study of non-Newtonian fluid flow. Hartnett and Kostic (6) assessed five friction factor correlations for the prediction of turbulent pressure drop of purely viscous power-law fluids in rectangular channels and circular tubes and concluded that (1) the Dodge-Metzner correlation gives the best agreement with the experimental results over the entire range of available power-law values (0.241 e n e 0.525), (2) the yoo correlation and the Irvine correlation are within (10% of the experimental data for n g 0.3, and (3) the Tam-Tiu correlation underpredicts experimental pipe flow data for n values between 0.2 and 0.5. Pinho and Whitelaw (7) investigated non-Newtonian fluid flow in a horizontal pipe and used a laser Doppler velocimeter to measure the mean axial velocity. The results quantified the delay in transition from laminar to turbulent flow caused by shear-thinning, the suppression of turbulent fluctuations, and the drag reduction at high Reynolds numbers. Hemeida (8) presented an equation for estimating the thickness of the laminar sublayer in turbulent pipe flow of pseudoplastic fluids and validated the friction factors against the reported correlations and field measurements. It was found that the hydraulically smooth pipes can be used to determine the pressure loss in the tested pipeline because the thickness of the laminar sublayer was greater than the average roughness height. Escudier and Presti (9) measured the mean velocity and velocity fluctuation levels using a laser Doppler anemometer for fully developed pipe flow of an aqueous solution of laponite and demonstrated that under turbulent flow conditions the logarithmic law was shifted upward by a small amount consistent with progressively reduced levels of drag reduction as the Reynolds number increased. Malin (10) performed numerical computations of fully developed turbulent flow of power-law fluids in smooth tubes and tested a modified version of the Lam-Bremhorst k-ε model. The predicted friction factor and mean velocity profile closely agreed with the measured data over a wide range of values for the power-law index and the generalized Reynolds number. Chilton and Stainsby (11) developed a numerical model for the simulation of laminar and turbulent flow of Herschel-Bulkley fluids and used the Launder-Sharma version of the low-Reynolds-number k-ε model to characterize the turbulent flow. Generally, the predicted pressure losses were lower than the experimental values within 15% error. Rudman et al. (12) conducted a direct numerical simulation (DNS) of the weakly turbulent flow of power-law and Herschel-Bulkley fluids in a small pipe and obtained very good agreement between the DNS results and experimental measurements. The DNS is a useful tool for performing fundamental research in turbulence because it numerically solves the Navier-Stokes equations without any turbulence model. However, the computational resources required by a DNS would exceed the capacity of the most powerful computers currently available (13). Wu and Chen (14) applied the standard k-ε model to simulate carboxymethyl cellulose (CMC) power-law fluid flow in a horizontal pipe, verified the axial velocities with the experimental data from Pinho and Whitelaw (7), and claimed that the overall prediction accuracies are acceptable. Wu (15) studied VOL. 44, NO. 23, 2010 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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gas and non-Newtonian fluid two-phase flow in a pipe at six total solids (TS) concentration levels and reported that the shear stress transport (SST) k-ω model could be used to simulate gas mixing in anaerobic digesters when TS e 5.4%. Even though both the CMC and manure slurry are nonNewtonian fluids, one question is whether a turbulence model that is applicable to one non-Newtonian fluid will apply to the others. The other question is whether the SST k-ω model should be recommended when solving singlephase flow. Therefore, the primary goal of this research was to evaluate 12 turbulence models on the basis of the criteria of frictional pressure drops in a horizontal pipe and propose the appropriate models that predict the single-phase nonNewtonian fluid flow at different TS levels. In addition, simulations of mixing by pumped circulation in a full-scale anaerobic digester were performed to investigate the effect of non-Newtonian rheology on the mixing energy, velocity gradient that impacts flocculation, optimum placement of discharge nozzles, and scaleup mechanism.
2. Development of the Numerical Model 2.1. Assumptions. (a) The critical Reynolds number for the laminar-turbulent flow transition in the pipe or the digester is 2000. (b) Liquid manure behaves as a pseudoplastic fluid for TS ranging from 2.5% to 12.1% at temperatures between 20 and 60 °C (16). (c) The digestion temperature is constant at 35 °C. (d) The model is single phase, in which the effects of any suspended sediments entrained within the manure and phase interaction due to the naturally produced biogas are negligible. 2.2. Reynolds-averaged Navier-Stokes (RANS) Turbulence Models. On the basis of the closure of the Reynolds stress term in the momentum equations, the RANS turbulence models are briefly described as follows. 2.2.1. k-ε Models. The k-ε models consist of the highReynolds-number k-ε (high-Re k-ε) and low-Reynoldsnumber k-ε (low-Re k-ε) models. For the standard k-ε model, the equations for k and ε are expressed as ∂ (Fk) + ∇·(Fv bk) ) ∇·[Γk∇k] + Gk - Yk + Sk ∂t
(1)
∂ (Fε) + ∇·(Fv bε) ) ∇·[Γε∇ε] + Gε - Yε + Sε ∂t
(2)
where k and ε are the turbulence kinetic energy and its dissipation rate, Γk and Γε are effective diffusivity of k and ε, Gk and Gε are the generation of k and ε, Yk and Yε are the dissipation of k and ε, and Sk and Sε are source terms. In the low-Re k-ε models, the turbulent viscosity for the standard k-ε model (µt ) FCµ k2/ε) and the transport equation for ε are modified as k2 µt ) FCµfµ ε
where fµ, f1, and f2 are the dumping functions that account for the effects of viscous damping and kinematic blocking. 2.2.2. k-ω Models. Similarly to the ε equation, the transport equation for ω is expressed as
9
(6)
where F ui′ uj′, Pi j, Πi j, Di j, and εi j represent the Reynolds stress, the stress-production, the pressure-strain, the diffusion, and the dissipation terms, respectively. 2.3. Non-Newtonian Rheology. The viscosity of manure slurry (η) and density (F) are expressed as (14) η ) Kγ˙ n-1eT0/T
(7)
F ) 0.0367(TS)3 - 2.38(TS)2 + 14.6(TS) + 1000
(8)
where K is the consistency coefficient, γ˙ is the shear rate, n is the power-law index, T0 is the reference temperature, T is the digestion temperature, and TS is the weight percentage of total solids in liquid manure. 2.4. Calculation of Frictional Pressure Drop. The frictional pressure drop per unit length in a pipe with a CFD simulation can be calculated by ∆Pf_CFD )
( )
τw /L A
(9)
where τw is the viscous force exerted by the wall on the fluid, A is the pipe cross-sectional area, and L is the pipe length. 2.5. CFD Simulation. The commercial CFD software packages Gambit 2.4.6 and Fluent 12.0 were used for meshing and solving the governing equations (17).
3. Theory of Correlation Analysis The Reynolds number for water flow in a pipe is defined as Re )
FuD µ
(10)
where u is the average velocity, D is the pipe diameter, and µ is the dynamic viscosity. The Fanning friction factor for turbulent flow is expressed as f )
0.079 (Re)0.25
(11)
The generalized Reynolds number for non-Newtonian fluid flow in a pipe is defined as (1) Reg )
Fu2-nDn k(0.75 + 0.25/n)n8n-1
(12)
Under turbulent flow conditions, several researchers have proposed the following correlations for predicting the friction factor. (a) Dodge and Metzner (2): 1 0.4 4 ) 0.75 log10(Regf (2-n)/2) - 1.2 n n √f
(13)
(b) Yoo (3): f ) 0.079n0.675Reg-0.25
(14)
2(2n/77n)1/(3n+1) (0.75 + 0.25/n)nReg1/(3n+1)
(15)
(5)
where ω is the specific dissipation rate, Γω is the effective diffusivity of ω, Gω is the generation of ω, Yω is the dissipation of ω, and Sω is the source term. 8990
D (Fui′uj′) ) Pij + Πij + Dij + εij Dt
(3)
∂ ε (Fε) + ∇·(Fv bε) ) ∇·[(µt/σε)∇ε] + (f1C1εGk - f2C2εFε) ∂t k (4)
∂ (Fω) + ∇·(Fv bω) ) ∇·[Γω∇ω] + Gω - Yω + Sω ∂t
2.2.3. Reynolds Stress Model. In the Reynolds stress model (RSM), the Reynolds stresses are directly computed. The second-order closure of the RSM can be expressed in a simple form as
ENVIRONMENTAL SCIENCE & TECHNOLOGY / VOL. 44, NO. 23, 2010
(c) Irvine (4): f )
TABLE 1. Frictional Pressure Drop from Correlation Analysisa K (Pa · s ) n ηmin (Pa · s) ηmax (Pa · s) F (kg/m3) u (m/s) Reg f_DM ∆Pf_Corr (Pa/m) f_Yoo ∆Pf_Corr (Pa/m) f_Irv ∆Pf_Corr (Pa/m) n
a
TS ) 2.5%
TS ) 5.4%
TS ) 7.5%
TS ) 9.1%
TS ) 12.1%
0.042 0.71 0.006 0.008 1000.36 2 6178 7.13 × 10-3 2854 7.07 × 10-3 2830 6.11 × 10-3 2446
0.192 0.562 0.01 0.03 1000.78 3 6316 6.1 × 10-3 5494 6.01 × 10-3 5410 4.65 × 10-3 4193
0.525 0.533 0.03 0.17 1001 4 4304 6.69 × 10-3 10715 6.38 × 10-3 10216 5.08 × 10-3 8131
1.052 0.467 0.07 0.29 1001.31 5 4865 5.9 × 10-3 14769 5.66 × 10-3 14163 4.19 × 10-3 10493
5.885 0.367 0.25 2.93 1001.73 6 2470 6.6 × 10-3 23801 5.7 × 10-3 20542 4.45 × 10-3 16046
The subscripts DM, Yoo, and Irv represent Dodge-Metzner, Irvine, and Yoo correlations, respectively.
TABLE 2. Frictional Pressure Drop from CFDa TS ) 0%
SKE RNG RKE SKO SST RSM Abid L-B L-S Y-S A-K-N C-H-C
TS ) 2.5%
TS ) 5.4%
∆Pf_CFD (Pa/m)
δ (%)
∆Pf_CFD (Pa/m)
δ_DM (%)
δ_Yoo (%)
δ_Irv (%)
∆Pf_CFD (Pa/m)
δ_DM (%)
δ_Yoo (%)
δ_Irv (%)
2651 2583 2657 2857 2280 2872 2705 2821 2482 2148 2678 2706
19 16 19 28 2.3 29 21 27 11 4 20 21
4799 4801 4799 3675 3981 3344 3252 3901 6073 3479 3409 2615
68 68 68 29 39 17 14 37 113 22 20 8
70 70 70 30 41 18 15 38 115 23 20 7
96 96 96 50 63 37 33 60 148 42 39 7
10908 10924 10908 8519 8997 7301 7525 8631 13992 8025 7904 5965
99 99 99 55 64 33 37 57 155 46 44 9
102 102 102 58 66 35 39 60 159 48 46 10
160 160 160 103 115 74 80 106 234 91 89 42
a The subscripts DM, Yoo, and Irv represent Dodge-Metzner, Irvine, and Yoo correlations, respectively. SKE ) standard k-ε, RNG ) RNG k-ε, RKE ) realizable k-ε, SKO ) standard k-ω, SST ) SST k-ω, RSM ) Reynolds stress model, L-B ) Lam-Bremhorst, L-S ) Launder-Sharma, Y-S ) Yang-Shih, A-K-N ) Abe-Kondoh-Nagano, and C-H-C ) Chang-Hsieh-Chen.
The frictional pressure drop with a correlation analysis can be calculated by ∆Pf_Corr )
2fFu2 D
(16)
Accordingly, an error indicator based on a correlation analysis can be calculated by δ)
|
|
∆Pf_CFD - ∆Pf_Corr × 100 ∆Pf_Corr
(17)
4. Results and Discussion 4.1. Examination of Turbulence Models. A horizontal pipe (0.02 m in diameter and 0.4 m in length) was used to examine 12 turbulence models that include (1) three high-Re k-ε models (standard k-ε, RNG (Renormalization Group) k-ε, and realizable k-ε), (2) six low-Re k-ε models (Abid, Lam-Bremhorst, Launder-Sharma, Yang-Shih, Abe-Kondoh-Nagano, and Chang-Hsieh-Chen), (3) two k-ω models (standard k-ω with shear flow corrections and SST k-ω with low-Re corrections), and (4) the RSM. These models have been incorporated into Fluent CFD software. To achieve mesh resolution near the pipe wall, a fine mesh having 52 836 cells was used (15). The enhanced wall treatments were set for the high-Re k-ε models and the RSM because this mesh was originally designed for the low-Re k-ε models. The inlet velocities of 2, 2, 3, 4, 5, and 6 m/s were set for TS ) 0%, 2.5%, 5.4%, 7.5%, 9.1%, and 12.1%, respectively, to ensure Re (or Reg)
> 2000. Using eqs 10 and 11, it can be determined that Re ) 40 000, f ) 5.59 × 10-3, and ∆Pf_Corr ) 2230 Pa/m at TS ) 0%. Table 1 shows ∆Pf_Corr from three different correlations for TS > 0, in which the rheological properties and densities were taken from Wu and Chen (14). If the criterion for selecting a turbulence model was set to be δ e 30% on the basis of any one correlation analysis (15), all the turbulence models overpredicted the frictional pressure drops at TS g 7.5%. Table 2 shows ∆Pf_CFD and δ at three TS levels, indicating that (1) all the turbulence models are applicable to TS ) 0% and (2) at TS ) 2.5% and 5.4% three priority turbulence models rank as the Chang-Hsieh-Chen version of low-Re k-ε, the Abid version of low-Re k-ε, and the RSM. If computing time is not an issue, these three models are recommended. The prediction discrepancy mainly results from the inherent limitations of using a model in an inappropriate application. The high-Re k-ε models are developed on the basis of the Boussinesq hypothesis that assumes locally isotropic turbulence, which might not be true for flows with heterogeneous turbulence. As an alternative approach to the Boussinesq hypothesis, the RSM solves transport equations for each Reynolds stress term along with an additional scaledetermining equation, which is usually superior for situations where anisotropy of turbulence has a dominant effect on the mean flows. The low-Re k-ε models are modified on the basis of the k-ε model to enable them to be used at low Reynolds numbers, which is appropriate for non-Newtonian fluid flow in this study where the Reynolds number is found VOL. 44, NO. 23, 2010 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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FIGURE 1. Velocity contours and profiles at the pipe exit: (C) coarse mesh of 18 156 cells for the standard k-ω model and (F) fine mesh of 52 836 cells for the Chang-Hsieh-Chen version of the low-Re k-ε model. Note that, in the velocity contours, blue represents areas that have low velocity while red represents areas of high velocity as indicated in the velocity contour bar, and the velocity magnitudes in the red area are greater than (or equal to) the maximum value specified in the bar. These representations apply to all the contour figures in this paper. center. Two nozzles were located in the same horizontal plane with 180° spacing (θ ) 180°) and each one discharging at an angle of 45° off the wall (β ) 45°). The effluent pipe consisted of two suction ports below the liquid surface, assuming that the area of each port is equal to that of one nozzle outlet. Table 4 shows five digester sizes, in which size 1 was used in all simulations unless noted. The mixing energy level can be calculated by (15)
FIGURE 2. Schematic diagram of the digester and velocity contours for MEL ) 5 W/m3. to be reduced significantly with an increase in TS. The k-ω models having low-Reynolds-number effects can be used with either a fine or a coarse near-wall mesh. Although two low-Re k-ε models and the RSM are superior to the other models when TS > 0%, they are computationally expensive since a low-Re k-ε model requires a very fine mesh near the wall while the RSM solves seven additional transport equations for three-dimensional flow. In comparison with the SST k-ω and three high-Re k-ε models, the standard k-ω model has a relatively low error when TS > 0%. To achieve a compromise between accuracy and computing time, the flow fields produced by the Chang-Hsieh-Chen version of the low-Re k-ε model were set as the reference and a coarse mesh having 18 156 cells without near-wall mesh resolution was generated in the pipe to examine the standard k-ω model. Figure 1 shows the comparison of flow patterns at the pipe exit using two turbulence models under two sets of meshes, from which it can be seen that the core flows are similar along the pipe center while the flows in the near-wall region are different because of the mesh density. From an engineering standpoint, the standard k-ω model can be used to simulate single-phase non-Newtonian fluid flow because of its acceptable accuracy and low computing cost. Hence, simulations hereafter were conducted using this model. 4.2. Effect of Rheology on the Mixing Energy. As shown in Figure 2a, a full-scale anaerobic digester constructed from a cylindrical tank with a conical bottom was used to investigate mixing by pumped circulation. The digester was equipped with a jet mixing loop comprised of two identical discharge nozzles at a low level and one effluent pipe at the 8992
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MEL ) E/V
(18)
E ) Fg∆HQ
(19)
where E is the power input, V is the working volume of the digester, ∆H is the hydrostatic head of the liquid, and Q is the discharging rate. Given MEL ) 5 W/m3 at TS ) 0% and d ) 0.079 m for the diameter of each nozzle, vin ) 6.46 m/s for the discharging velocity of each nozzle can be determined from eqs 18 and 19. Because of insignificant density differences within TS e 5.4%, the discharging velocity calculated from TS ) 0% was applied to the other two TS levels under the same mixing energy. To perform quantitative analysis, the mixing intensity was defined as the average velocity throughout the whole j and the turbulence intensity was defined as digester tank (v) (ui′2)1/2/vref, where ui′ are the turbulent velocity fluctuations and νref ) 1 m/s. Parts b-d of Figure 2 show the velocity contours along the central plane from the front view at three TS levels, from which it can be observed that all the flow patterns are symmetrical about the effluent pipe and that the mixing intensity decreases with an increase in TS. Taking TS ) 0% as an example, Figure 3 shows plots of the velocity magnitude and turbulence intensity across the tank diameter at two depths. At z ) 1.2 m the velocity and turbulence intensity increase from the side wall to the nozzles, reach their peaks at the nozzle exits, and then decline toward the tank center, while at z ) 7.2 m relatively strong turbulence can be observed from both the near-wall and effluent zones. These predictions were generally consistent with the results from a similar CFD study in which both inlets and outlets were set at the tank wall (18). To characterize the mixing intensity with respect to the mixing energy, extensive simulation work was done to obtain j the v-MEL function as
{
c ) 0.0415 TS ) 0% vj ) c(MEL) c ) 0.0388 TS ) 2.5% c ) 0.0365 TS ) 5.4%
(20)
As indicated by the U.S. EPA, MEL ) 5-8 W/m3 is recommended for effective mixing in anaerobic digesters by pumped circulation (19). However, this guideline does not address the type of anaerobic digester being used and the fluid being mixed. In this study, it was presumed that MEL ) 5-8 W/m3 could be applied to mix water in a cylindrical digester, and then vj ) 0.208-0.332 m/s at TS ) 0% could be
predicted on the basis of this assumption. Through inversely solving eq 20, MEL ) 5.36-8.55 and 5.7-9.1 W/m3 were proposed to be the ranges of mixing energy levels for acquiring vj ) 0.208-0.332 m/s at TS ) 2.5% and 5.4%, respectively. Thus, the rheological properties of a fluid being mixed should be taken into consideration when using the effective mixing standards recommended by the U.S. EPA. 4.3. Characterization of the Velocity Gradient. The average velocity gradient (G) is defined as (20) G)
E µV
(21)
The local velocity gradient is defined as (20) GL )
νε
(22)
where ν is the kinematic viscosity. The breakup number is defined as (20) B)
F J
(23)
where F is the rupture force and J is the aggregate strength. The floc breaks up when B > 1, while its size is maintained or increased when B < 1. The rupture force can be approximated by F ≈ σdf2
(24)
where σ is the hydrodynamic stress (σ ) µ GL in the viscous subrange, while σ ) F C2(ε df)2/3 in the inertial subrange) and df is the floc diameter. Substituting ω ) ε/(β*k) in the k-ω model developed by Wilcox (21) and ν ) η/F for non-Newtonian fluids in eqs 22 and 23 yields
Fωβ*k η
(25)
C1η√Fωβ*k/ηdf2 J
(26)
FC2(Fωβ*k/η)2/3df8/3 J
(27)
GL )
Bvsr ) Bisr )
where β* ) 0.09, C1 ) 5π/8, C2 ) 0.7, Bvsr is the breakup number in the viscous subrange, and Bisr is the breakup number in the inertial subrange. Equations 25-27 can be solved using three custom field functions in the Fluent 12.0 software. Figure 4 shows the contours of the velocity gradient for MEL ) 5 W/m3 at three TS levels, from which it can be shown that the maximum GL occurs at the nozzle exits while the minimum GL occurs at the tank bottom. Theoretically, eq 21 does not apply to a non-Newtonian fluid in which the viscosity (η) varies with the shear rate (γ˙ ) unless an equivalent µ for this fluid is available. Given MEL ) 5 W/m3 and µ ) 0.001 Pa · s at TS ) 0%, G ) 70.7 s-1 can be acquired by solving eq 21. However, checking the GL distributions shows that the percentages of volume with respect to GL are 84.5%, 7.9%, and 7.6% for GL < 10 s-1, 10 s-1 e GL < 70 s-1, and GL g 70 s-1, respectively. The average GL can be predicted as 3.44 s-1 via the integration of GL over the whole computational domain, demonstrating that only a small percentage of volume has a GL over 70 s-1. Clearly, the velocity gradients in the digester are nonuniform. These results were in qualitative agreement with the findings from related research (22). Therefore, the concept for the local velocity gradient is more reasonable than the traditional definition of the G value that characterizes a complex flow field with a single number. To investigate the effect of turbulence on flocculation, two points in the digestion tank were specified as P1 (5, 0, 1) and P2 (5.5, -0.5, 1.2). Note that the Cartesian coordinate system was fixed at the bottom center with metric units. Given df ) 100 µm and J ) 5 × 10 -9 N for a floc (20), Bvsr
FIGURE 3. Mean velocity and turbulence intensity for MEL ) 5 W/m3 at TS ) 0%.
FIGURE 4. Contours of the velocity gradient for MEL ) 5 W/m3. VOL. 44, NO. 23, 2010 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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TABLE 3. Mixing Intensity versus Orientation of the Nozzles v¯ (m/s) versus θ at β ) 45° TS (%) 0 2.5 5.4
45°
90°
135°
180°
v¯ (m/s) versus β at θ ) 180° 15°
30°
60°
TABLE 4. Design Parameters and Mixing Intensity for Five Digestersa
75°
0.200 0.204 0.210 0.208 0.209 0.209 0.197 0.150 0.187 0.189 0.199 0.195 0.201 0.194 0.188 0.145 0.175 0.177 0.180 0.181 0.179 0.180 0.180 0.138
and Bisr at P1 can be predicted as 0.13 and 0.003, respectively, indicating that mixing at P1, which is near the conical wall, favors the floc growth. Similary, Bvsr ) 13.86 and Bisr ) 1.71 can be achieved at P2, indicating that mixing at P2, which is near the right discharge nozzle, causes the floc breakage. In physics either Bvsr or Bisr at a point is true depending on into which subrange this point falls. It should be pointed out that rheological and turbulent effects on solids flocculation and breakup are among the most challenging topics in anaerobic digestion research, which requires experimental work as an aid in further investigation. 4.4. Optimum Placement of Discharge Nozzles. To test the effect of the spacing between two nozzles (θ) on the mixing, simulations of flow fields were carried out at three TS levels for θ ) 45°, 90°, and 135°, in which the discharging angle for each nozzle (β) was kept at β ) 45°. Similarly, sensitivity analysis of the discharging angle ranging from 15° to 75° while keeping θ ) 180° was conducted. Table 3 shows that (1) the mixing intensity changes insignificantly with an increase of θ from 45° to 180° and (2) the mixing intensity remains almost unchanged over the range of β ) 15-30° while it decreases with an increase of β from 60° to 75°. Taking TS ) 5.4% as an example, Figure 5 shows that the flow patterns for θ ) 180° are more uniform than those for θ < 180° and that an increase in β results in moving the intensive mixing from the near-wall region to the central region. In terms of the mixing intensity and uniformity, mixing design by θ ) 180° and β ) 45° for two discharge nozzles is recommended.
3
V (m ) DT (m) H (m) h (m) C (m) de (m) d (m) v¯ at TS ) 0% (m/s) v¯ at TS ) 2.5% (m/s) v¯ at TS ) 5.4% (m/s)
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size 2
size 3
size 4
size 5
792 12 6.7 0.9 1.2 0.158 0.079 0.208 0.195 0.181
1188 13.74 7.672 1.031 1.374 0.182 0.09 0.208 0.196 0.182
1584 15.12 8.442 1.134 1.512 0.199 0.099 0.208 0.196 0.183
1980 16.29 9.095 1.222 1.629 0.214 0.107 0.208 0.197 0.184
2376 17.31 9.665 1.298 1.731 0.228 0.114 0.208 0.198 0.185
a V ) working volume of the digester, DT ) tank diameter, H ) cylindrical height, h ) conical height, C ) clearance of each nozzle, de ) diameter of the effluent pipe, and d ) diameter of each nozzle.
4.5. Scaleup Mechanism. The purpose of scaleup is to design a large mixing system that will achieve the same mixing characteristics as in a small one. In this study, the scaleup with geometric and mixing energy similarities was examined. The working volumes for four digesters were assumed to be V2 ) 1.5V1, V3 ) 2V1, V4 ) 2.5V1, and V5 ) 3V1, in which the subscripts 1, 2, 3, 4, and 5 represent size 1, size 2, size 3, size 4, and size 5, respectively. If the tank diameter (DT) was chosen as the dimension basis, then H/DT, h/DT, C/DT, de/DT, and d/DT in each digester were kept constant. Under the same mixing energy level, the discharging velocity of each nozzle at any digester size can be predicted as vin ) 6.46 m/s by solving eqs 18 and 19. As shown in the last three rows of Table 4, the mixing intensity stays unchanged at TS ) 0% while it increases slightly with an increase in the digester size at TS ) 2.5% and 5.4%. If the numerical errors are taken into account, the flow fields at any digester size remain much
FIGURE 5. Velocity contours versus placement of discharge nozzles at TS ) 5.4%. 8994
size 1
the same with a constant mixing energy level and a specified TS level. Therefore, the design of scaleup digesters with pumped circulation should follow the similarity rules of geometry and mixing energy.
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(11) Chilton, R. A.; Stainsby, R. Pressure loss equations for laminar and turbulent non-Newtonian pipe flow. J. Hydraul. Eng. 1998, 124 (5), 522–529. (12) Rudman, M.; Blackburn, H. M.; Graham, L. J. W.; Pullum, L. Turbulent pipe flow of shear-thinning fluids. J. Non-Newtonian Fluid Mech. 2004, 118, 33–48. (13) Jiang, X.; Lai, C. Numerical Techniques for Direct and LargeEddy Simulations; CRC Press: London, U.K., 2009. (14) Wu, B.; Chen, S. CFD simulation of non-Newtonian fluid flow in anaerobic digesters. Biotechnol. Bioeng. 2008, 99 (3), 700–711. (15) Wu, B. CFD simulation of gas and non-Newtonian fluid twophase flow in anaerobic digesters. Water Res. 2010, 44 (13), 3861–3874. (16) Achkari-Begdouri, A.; Goodrich, P. R. Rheological properties of Moroccan dairy cattle manure. Bioresour. Technol. 1992, 40, 149–156. (17) ANSYS-Fluent Inc. Fluent 12.0; Lebanon, NH, 2008. (18) Meroney, R. N. CFD simulation of mechanical draft tube mixing in anaerobic digester tanks. Water Res. 2009, 43 (4), 1040–1050. (19) U.S. Environmental Protection Agency. Process Design Manual for Sludge Treatment and Disposal; EPA/625/1-79/011; EPA: Cincinnati, OH, 1979. (20) Coufort, C.; Bouyer, D.; Line, A. Flocculation related to local hydrodynamics in a Taylor-Couette reactor and in a jar. Chem. Eng. Sci. 2005, 60, 2179–2192. (21) Wilcox, D. C. Reassessment of the scale-determining equations for advanced turbulence models. AIAA J. 1988, 26 (11), 1299– 1310. (22) Bridgeman, J.; Jefferson, B.; Parsons, S. A. The development and application of CFD models for water treatment flocculators. Adv. Eng. Software 2010, 41, 99–109.
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