ARTICLE pubs.acs.org/est
Can a Stepwise Steady Flow Computational Fluid Dynamics Model Reproduce Unsteady Particulate Matter Separation for Common Unit Operations? Subbu-Srikanth Pathapati and John J. Sansalone* Environmental Engineering Sciences, University of Florida, 110 Black Hall, Gainesville, Florida, 32611-6450 United States
bS Supporting Information ABSTRACT: Computational fluid dynamics (CFD) is emerging as a model for resolving the fate of particulate matter (PM) by unit operations subject to rainfallrunoff loadings. However, compared to steady flow CFD models, there are greater computational requirements for unsteady hydrodynamics and PM loading models. Therefore this study examines if integrating a stepwise steady flow CFD model can reproduce PM separation by common unit operations loaded by unsteady flow and PM loadings, thereby reducing computational effort. Utilizing monitored unit operation data from unsteady events as a metric, this study compares the two CFD modeling approaches for a hydrodynamic separator (HS), a primary clarifier (PC) tank, and a volumetric clarifying filtration system (VCF). Results indicate that while unsteady CFD models reproduce PM separation of each unit operation, stepwise steady CFD models result in significant deviation for HS and PC models as compared to monitored data; overestimating the physical size requirements of each unit required to reproduce monitored PM separation results. In contrast, the stepwise steady flow approach reproduces PM separation by the VCF, a combined gravitational sedimentation and media filtration unit operation that provides attenuation of turbulent energy and flow velocity.
’ INTRODUCTION Urban runoff is a complex mixture of heterodisperse particulate matter (PM) and solutes1 that dynamically partition with PM. Management of PM is challenged by the coupling of heterodisperse PM transported in response to unsteady hydrologic phenomena. While larger watershed-level controls exist, decentralized urban catchment-level control by smaller footprint unit operations are numerically much more frequent. Over the last several decades monitoring of PM separation by these unit operations, commonly a retention/detention basin or tank, has been based on the lumped gravimetric index of PM, total suspended solids (TSS), and event mean concentrations (EMCs) of TSS.25 Over the same period significant urban runoff modeling advances have been hydrologically based. For example, models such as the Stormwater Management Model (SWMM) provide robust representation of unsteady intraevent flows and continuous simulations. Despite unsteady runoff flows and concentrations, models have gravitated toward available data as EMCs of lumped indices such as TSS.6,7 Such watershed models can require extensive calibration with lumped PM indices such as TSS.8 Beyond TSS and EMCs, detailed monitoring, analysis, and modeling for intraevent transport and fate of PM by unit operations is less common.2,3 Lack of detailed flow monitoring and PM analysis (such as particle size distributions, PSDs) can introduce significant errors. For example, studies of wet basins illustrate that without detailed data, mean errors for effluent PM range from 40 to 50%.9,10 Studies indicate that accurate hydraulic and PSD representation are critical r 2011 American Chemical Society
to predict behavior.11 It is hypothesized that accurate representation of flow and PSDs is critical for smaller unit operations; for example tanks, vaults, clarifiers, and HS units with a smaller footprint and lower hydraulic attenuation. Regulatory programs requiring interim certification for unit operations require physical modeling with steady flows and PSDs as a precursor to uncontrolled field testing.4,5 Although such programs do not yet require numerical modeling, controlled physical modeling permits mechanistic and numerical model development. With respect to unsteady flows and PSDs, representative monitoring and numerical modeling of unit operations is more complex. CFD is emerging as a tool to resolve the coupling of hydraulics, PM transport, and PM discretization, albeit as applied for steady flow and loadings. For example, CFD reproduces hydrodynamic separator (HS) and a filter system behavior under steady flow and constant PSDs.12,13 While CFD can now extend to transient flows and variable PSDs14,15 the consequences are higher computational effort and error. A study of unsteady loadings concluded that mean flow rate14 as a lumped flow parameter misrepresents PM separation of a unit operation compared to an unsteady CFD model validated with intraevent monitoring. For steady flows, heterodisperse PSDs have been Received: October 23, 2010 Accepted: May 13, 2011 Revised: May 9, 2011 Published: June 06, 2011 5605
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Exceptions are noted to illustrate the impact of further increasing the discretization. The primary objective of this study is to investigate if an integrated stepwise steady approach can reproduce the monitored unsteady PM separation by three common unit operations applied at the urban catchment level. These small footprint unit operations are a HS, a primary clarifier (PC) tank, and a VCF combining volumetric clarification and filtration.
’ MONITORING METHODOLOGY Isometric views of the three unit operations are illustrated in Figure 1. Each unit is sized based on catchment hydraulic loadings. The VCF consists of five gravity-driven radial cartridge filters (RCF) within a sedimentation vault. Each RCF contains approximately 49.8 kg of monodisperse Al-oxide infused granular media that is porous with a median diameter of 3.5 mm (0.8 mm and specific gravity of 2.35 ( 0.01. The total (bed þ media) porosity of each RCF is 0.71 ( 0.041. In contrast the PC tank functions solely as a baffled type I settling unit operation. The HS consists of two concentric chambers separated by a screen. PM separation by the HS is predominantly due to gravitational separation and to a lesser extent by size-separation screening. As physical models loaded by unsteady runoff events, the VCF, PC, and HS are evaluated in the same urban transportation land use watershed in Baton Rouge, LA. Detailed results regarding the watershed, hydrology, PM loads, and water chemistry are available elsewhere.17 Eight, six, and four events for the VCF, the HS, and the PC, respectively, are modeled. The events each provide required influent and effluent flow and PM data for CFD model comparisons. Table 1 summarizes event hydrology. Monitoring is based on manual full cross-sectional flow samples for the entire event duration. The median numbers of monitoring points are 13, 13, and 11 for the VCF, PC, and HS, respectively; within the range of sample number requirements for evaluating event-based PM loads.18,19 To examine the potential for error reduction of the stepwise steady CFD model this discretization is extended to 36 points through PSD interpolation. Detailed information of the effect of extended discretization is provided in Supporting Information. Previous research has also recommended that the entire PSD is required to design and examine physical unit operations.17,20 The measured PSD resolution ranges from 0.02 to 2000 μm, in 100 logarithmic size increments. A mass balance error limitation is set for each event as follows. Minf ðMUOP þ Mef f Þ Mass Balance Error ðMBEÞ ¼ Minf 100 e ( 10%
ð1Þ
Figure 1. Isometric views of the geometry of (A) volumetric clarifying filtration system (VCF), (B) primary clarifier, and (C) hydrodynamic separator (HS).
Minf is the PM influent mass, MUOP is the PM mass separated, and Meff is the PM effluent mass.
discretized in order to balance computational effort and model error.16 This leads to the question of whether an unsteady runoff event can be resolved as a series of steady flows within CFD in order to predict PM separation behavior with less computational effort. Therefore, to reproduce the monitored unsteady PM separation response by unit operations this study tests if unsteady runoff loadings can be discretized into a series of steady flow steps. Discretized flow steps each separately resolved in CFD are taken at intraevent monitoring points where PSDs are measured.
’ MODELING METHODOLOGY A three-dimensional approach is required to characterize the continuous phase hydrodynamics and PM dynamics. The NavierStokes equations for incompressible flows12,13,21 are applied to each unit operation and are not reproduced herein for brevity. The standard kε model is utilized for k (turbulent kinetic energy) and ε (rate of dissipation of k) to model the continuous phase flow.1215 Isotropy of Reynolds stresses is assumed for the HS, PC, and VCF and standard values of model constants22 are used. The computational geometry is discretized 5606
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Table 1. Hydrologic Indices Across Storm Events for VCF, PC, and HSa Drunoff storm event
21 April 2006
min
Qinf-max L/s
Qinf-avg
Vin
L/s
L
volumetric clarifying filtration (VCF) 55 13.3 1.2 2927
NTotal influent effluent
13
14
29 April 2006 06 May 2006
170 36
25.4 0.3
5.1 0.1
48306 495
18 11
18 13
07 May 2006
50
9.1
2.5
3852
11
12
27 May 2006
43
6.5
1.8
2628
10
12
28 May 2006
40
7.4
1.5
2096
10
12
16 June 2006
70
14.6
2.1
9938
11
13
19 June 2006
32
4.3
1.0
1816
11
13
05 June 2005
primary clarification (PC) 51 9.4 1.9 5856
21 April 2006 29 April 2006 05 July2006
14 March 2004
14
13
55 170
13.3 25.4
1.2 5.1
2927 48306
13 18
14 18
64
7.9
1.0
3838
14
14
hydrodynamic separation (HS) 408 6.4 1 24076
17
13
199
1.8
0.6
7288
16
15
20 August 2004
41
17.5
5.1
12286
15
14
14 October 2004
169
0.6
0.1
1672
14
13
51 71
9.4 13.8
1.9 3.6
5856 15117
14 16
13 15
21 August 2005
106
17.3
7.9
50002
16
15
03 October 2005
15
12.1
3.1
2615
9
6
24 April 2004
05 June 2005 30 June 2005
a Drunoff, Qinf-max, Qinf-avg, Vin, and NTotal are the duration of runoff, peak influent flow rate, mean influent flow rate, total influent volume, and total sample number (monitoring points for PSD and PM), respectively.
using an unstructured mesh with tetrahedral elements. Grid independence is achieved for the VCF, PC, and HS at 5.18, 3.6, and 1.96 million computational cells, respectively. This study utilizes the finite volume method (FVM) for CFD. A cellcentered scheme is used in the geometric discretization and is a second-order upwind scheme for computing values at cell faces. Pressurevelocity coupling is accounted for by the SIMPLE (Semi-Implicit Method for Pressure Linked Equations) algorithm.23 The criterion for iterative convergence is 1 103.24 The HS static screen is modeled as a porous plate by a momentum sink term in the fluid flow equations.13,14 This sink term is composed of a viscous loss term and an inertial loss term. The static screen is modeled as a surficial filter, with the sink term expressed by the following equation. μ 1 vi þ C2i Fvmag vi ð2Þ Si ¼ R 2 Si is the source term for the ith momentum equation, R is the permeability, C2i is the inertial resistance coefficient, vi is the velocity in the ith momentum equation, vmag is the velocity in the computational cells of the RCF component of the VCF and the screen component of the HS. The RCF is modeled with a macroscopic approach utilizing head loss parameters such as effective porosity (η), inertial, and viscous resistance coefficients.12,15 Parameters are calculated assuming media homogeneity
for pore size distribution and media diameter. This is reasonably assumed as the media is pluviated into a RCF and is monodisperse. The Ergun equation25 for filter bed head loss is applicable across all flow regimes and is expressed as follows. Δp k0 ðLe =LÞ2 μ ð1 ηÞ2 am 2 ð1 ηÞ am vs 2 ¼ νs þ k2 L Fg η3 η3 Vm Vm g ð3Þ am 6 ¼ Φdm Vm
ð4Þ
In these expressions, Δp is the pressure drop across the media, L is the nominal flow path length of the packed bed, μ is the fluid viscosity, dm is media particle diameter, η is the total porosity of the packed bed, νs is the superficial velocity through the packed bed and F is fluid density, am/Vm is media surface area per unit of media volume, k0 is a shape factor, k2 is a dimensionless Ergun constant, Φ is the sphericity factor (75 μm) largely separated by the VCF and PC. Figure 2a summarizes the cumulative frequency distributions of modeled fluid velocities in the VCF, PC, and HS. These velocity distributions are statistically significantly different (p e 0.05), with the VCF having the lowest velocity magnitude distribution and the HS having the highest velocity magnitude distribution of all the units. The increased resistance to flow is due to the RCFs with the media diameter and packing yielding Reynolds numbers ranging from predominantly laminar into the transitional regime. This is further illustrated by the lower fluid velocities in the filter domain of the VCF, shown as cumulative frequency distributions in Figure 2b and instantaneous head loss up to approximately 150 mm across the RCF. The VCF also provides flow attenuation from influent to effluent, in contrast to the HS. ’ PROBABILITIES OF PM SEPARATION BY THE VCF, PM, AND HS Figure 3 illustrates the cumulative probabilities for PM mass separation by the VCF, PC, and HS as a function of PM diameter and influent flow rate. The maximum monitored flow rate for
each unit is chosen as the upper limit. At high flow rates, the VCF and PC completely separate PM larger than 300 μm, while the HS separates PM larger than 2000 μm. At a low flow rate of 0.1 L/s the VCF and PC separate all PM larger than 30 μm. Figure 3 also illustrates the probabilities of separating a PM size fraction at a chosen flow rate. A gamma probability density function represents PM separation as a function of flow rate, Q, and particle diameter, symbolized as “x”. γ 1 ð x Þ x e β β ð15Þ f ðxÞ ¼ ðβÞΓðγÞ γ and β are shape and scaling factor, respectively, and F(x) is the cumulative gamma distribution. Z x FðxÞ ¼ f ðxÞdx ð16Þ 0
Variation of γ and β with flow is plotted in the right-hand column of Figure 3. Heterodispersivity of a PSD has been shown to be proportional to γ and inversely proportional to β.16 At a constant γ, an increasing β indicates a coarser PSD while for a constant β; an increasing γ indicates a finer PSD. For the VCF there is an increase in β with increasing flow, while γ remains approximately constant. Trends in γ and β with flow rate indicate that while the VCF is able to separate a large range of PM irrespective of flow rate. In contrast to the VCF, PM separation for the PC and HS is primarily coarse PM as illustrated by the greater variability of γ for the HS.
’ STEPWISE STEADY VERSUS UNSTEADY MODEL PREDICTIONS All unsteady CFD model predictions are based on integrating model results at each monitored PSD and flow rate in time. In contrast, for the stepwise steady flow CFD model, results at each discretized steady flow level are flow-weighted across the unsteady runoff event. Each method is carried out for the entire runoff duration utilizing monitored PSDs. RMSE and RPD are used to quantify model error for the fully unsteady and stepwise steady CFD methods as compared to measured PM separation. Such model error can impact sizing criteria for UOPs. One reason for the proliferation of manufactured UOPs such as the HS and the filtration system is arguably their requirement of less 5608
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Figure 3. Probabilities of PM separation by the VCF, PC, and HS plotted as response surfaces (left). Parameters of gamma distributions plotted as a function of steady flow rates (right).
land area.28 In addition, since PM separation is a function of the unit’s appropriate form of surface area (SA), surface area is utilized as a primary design parameter and unit sizing index that is
impacted by model error. For the HS and PC the appropriate SA index is the actual surface overflow area of each unit combined with Newton’s Law for discrete Type I particle settling which is 5609
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Table 2. CFD Model Comparisons to Measured Data for PM Mass for the VCF, PC, and HS measured data
max stepwise steady, Q = ∑t=t Qt t=0
unsteady, Q = f(t)
PM Load rainfall- runoff BMP and separation mechanisms
event date
influent effluent (g)
CFD modeled effluent PM load (g)
(g)
CFD modeled effluent PM RMSE
load (g)
RMSE
VCF ( discrete settling, granular- media based
21 April 2006
4161
119
110
1.34
102
1.2
radial filtration)
29 April 2006
10466
1678
1676
3.6
1822
5.1
06 May 2006
172
13
15
2.9
16
4
07 May 2006
1630
132
120
2.6
159
3.1
27 May 2006
1219
93
80
3.1
104
5.1
28 May 2006
320
39
57
6.9
46
2.5
16 June 2006 19 June 2006
5363 541
401 40
413 44
2.7 2.1
415 47
2.4 3.1
primary clarifier (discrete settling)
HS (discrete settling, screening, vortex-driven separation)
05 June 2005
1581
743
819
4.5
939
24.9
21 April 2006
4161
1129
230
14.3
431
40.3
29 April 2006
10466
3875
2395
6.5
4472
25.6
05 July 2006
1065
370
433
4.1
489
6.8
14 March 2004
4949
2593
3174
3.43
8669
35.8
24 April 2004
3219
976
1141
2.6
1946
15.2
20 August 2004 14 October 2004
10592 544
4314 268
5016 309
2.49 2.38
12519 487
29.1 12.5
05 June 2005
4758
1582
1841
2.51
3122
14.9
30 June 2005
4044
2495
2946
2.77
5249
16.9
21 August 2005
8838
4884
5886
3.14
15195
32.3
03 October 2005
738
382
468
3.46
724
13.7
appropriate given the coarser, heterodisperse PSD loadings and residence times that are nominally less than an hour and around the peak of the hydrograph on the order of several minutes. For the VCF the appropriate SA index is the radial SA of the filter for surface loading rate determination. The radial filters control the PM separation behavior of the VCF at a given surface loading rate. The error generated by CFD modeling expresses the difference in PM separation from the measured PM separation. This difference is utilized to determine the additional SA required as a result of model error while achieving the same level of PM separation from the monitored physical model results. Specifically, a mean and standard deviation of PM separation from each unit was utilized from the monitoring campaign. The SA increase that is generated from increasing model error (as RMSE) and illustrated as a factor of the existing unit SA is determined through a linear correlation shown in Figure S3 of the Supporting Information. The rationale behind creating this relationship, shown in the SI Figure S3 is to provide an estimation of the implications of modeling error on UOP design. For example, a CFD model of the VCF generating an RMSE of 10 would suggest a design sizing requirement of 4 additional filters and 2.75 times more SA to maintain the same level of PM separation as determined from the monitored physical model results. Similarly, this RMSE of 10 results in approximately 2.5 and 3 times larger SA requirements for the PC and HS, respectively. Table 2 summarizes event-based monitored and CFD modeled effluent PM mass. For PM mass the error arising from the stepwise steady (quasi-steady) approach is lowest for the VCF and highest for the HS. At low flows, the stepwise steady model deviation from the fully unsteady CFD model is lower than at high flows. The role of peak flow rate on the stepwise steady
model error is pronounced for the PC and HS, in contrast to the VCF, where RMSE is low irrespective of flow. Figure S4 in the Supporting Information illustrates these results. Utilizing monitored PSDs and flows, Figure 4 compares unsteady and stepwise steady models to monitored effluent PM mass. The filled line plots represent the influent hydrograph, open circles represent measured effluent PM mass, and the solid lines and dashed lines represent effluent PM mass predicted by the stepwise steady and full unsteady CFD models respectively. These intraevent results illustrate that the stepwise steady model for the VCF is relatively representative of monitored PM across each event. In comparison, there is increasing intraevent deviation from monitored PM results using the stepwise steady model for the PC and HS, respectively. Only at lower flow rates early in an event did the stepwise model represent monitored PM load. These results are corroborated with the velocity profiles in the three units. Given that the PC has the same geometry as the VCF vault, the high RMSE values of the PC illustrate the dominant role of a filter to dissipate turbulent energy and reduce flow velocity down-gradient of sedimentation as shown in Figure 2b. In addition, the errors are progressively larger for the PC and HS as compared to the VCF that combines sedimentation and filtration. Computational time is examined using a workstation equipped with dual quad-core 2.6 GHz processors and 32 GB of random access memory. For stepwise steady simulations, run times are 1.5 ( 0.5, 1.7 ( 0.5, and 2.5 ( 0.5 h for the HS, PC, and VCF, respectively, as compared to 2.5 ( 0.5, 2.5 ( 0.5, and 3 ( 0.5 days for fully unsteady simulations. Having resolved the hydrodynamics and PM separation of a unit operation for a chosen number of stepwise steady flows, these resolved steps can 5610
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Figure 4. Comparison of measured effluent PM mass (as SSC) with fully unsteady and stepwise steady (quasi-steady) CFD models for the VCF, PC, and HS. Results are presented on a cumulative basis. Open circles represent measured effluent PM mass. Solid and dashed lines represent effluent PM mass predicted by the stepwise steady and fully unsteady models, respectively. The filled line plot represents the influent hydrograph.
be applied across a series of unsteady events. In contrast, each unsteady simulation requires resolution of variable flows and the DPM across the entire duration of each discrete rainfallrunoff event. Therefore, the unsteady computational time for the VCF, for example, is a factor of 16 times greater than for the stepwise steady CFD model.
’ CONVERGENCE OF STEPWISE STEADY AND UNSTEADY METHODS Figure 5 depicts model error for PM separation as a function of integrating over an increasing number of consecutive stepwise steady flow steps. Error at each discretization step is calculated relative to monitored PM separation. With the VCF, the median RMSE decreases to 4 as the number of discretization steps approaches the number of monitoring points. Error generation
is a function of hydrology and approaches an asymptote with increasing time past the peak flow rate of each runoff event. This is consistent with research14 reporting that the peak flow rate is likely to generate less error than a mean flow rate. Convergence of the stepwise-steady and fully unsteady approaches is not observed for either the PC or HS even at increasing discretization up to 36 as shown in Figure S5. As the steady steps are increased from 8 to 12 the error approaches differing asymptotic limits depending on the unit. PC and HS errors remain large even at higher levels of discretization suggesting inapplicability of the stepwise steady approach for these units. The stepwise steady approach is favored for the VCF which provides dissipation of turbulent energy and flow velocity compared to the PC and HS. An extension of stepwise steady pilot-scale test results is not directly portable to real-time storm events for units with complex 5611
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Figure 5. Median and variation of the stepwise steady (quasi-steady) model error (RMSE) with increasing number of monitoring points (for PM and PSDs) for VCF, PC, and HS. For comparison, median values for the fully unsteady model are 3.1, 5.5, and 2.6 for VCF, PC, and HS respectively. NTotal is the total number of monitoring points. The lower right quartile box plot is the variation of error across the influent flow rate range.
turbulent hydraulics such as HS-type units and high rate small clarifiers without significant attenuation of flow velocities. However, the stepwise steady approach, and subsequently, pilot-scale testing can be a viable alternative to monitoring for filtrationbased units or larger basins providing hydraulic attenuation. While the fully unsteady CFD model provides the most accurate representation for unit operations, the stepwise steady approach for low-turbulence filtration can reproduce measured PM separation at significantly lower computational time. Whether the PM separation model is as basic as an overflow rate model18 or as complex as a CFD model, representative input PSD and flow data are required.
’ ASSOCIATED CONTENT
bS
Supporting Information. Additional figures, text and references. This information is available free of charge via the Internet at http://pubs.acs.org/.
’ AUTHOR INFORMATION Corresponding Author
*E-mail: jsansal@ufl.edu; phone: þ352-846-0176; fax: þ352392-3076.
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