Environ. Sci. Technol. 2009, 43, 8220–8226
Discrete Phase Model Representation of Particulate Matter (PM) for Simulating PM Separation by Hydrodynamic Unit Operations JOSHUA A. DICKENSON* AND JOHN J. SANSALONE Environmental Engineering and Sciences, University of Florida, 217 Black Hall, P.O. Box 116450, Gainesville, Florida 32611
Received May 25, 2009. Revised manuscript received September 4, 2009. Accepted September 17, 2009.
Modeling the separation of dilute particulate matter (PM) has been a topic of interest since the introduction of unit operations for clarification of rainfall-runoff. One consistent yet controversial issue is the representation of PM and PM separation mechanisms for treatment. While Newton’s Law and surface overflow rate were utilized, many historical models represented PM as a lumped gravimetric index largely out of economy and lack of particle analysis methods. As a result such models did not provide information about particle fate in or through a unit operation. In this study, PM discrete phase modeling (DPM) and computational fluid dynamics (CFD) are applied to model PM fate as a function of particle size and flow rate in two common types of hydrodynamic separator (HS) units. The study examines the discretization requirements (as a discretization number, DN) and errors for particle size distributions (PSDs) that range from the common heterodisperse to a monodisperse PSD. PSDs are categorized based on granulometric indices. Results focus on ensuring modeling accuracy while examining the role of size dispersivity and overall PM fineness on DN requirements. The fate of common heterodisperse PSDs is accurately predicted for a DN of 16, whereas a single particle size index, commonly the d50m, is limited to monodisperse PSDs in order to achieve similar accuracy.
Introduction Particulatematter(PM)isacommonpollutantinrainfall-runoff (1) and surface waters (2). As historically practiced, gravimetric indices such as total suspended solids (TSS) and suspended sediment concentration (SSC) do not provide particle size distribution (PSD) representation. However, PM discretization methods allow examination of chemical and biological interactions with PSDs, for example with metals (3). Silt- and clay-size PM provide habitat and a protective matrix for microbes (4) and is a mobile substrate that provides disinfection resistance (5). As a result, PM clarification is common for treatment of stormwater (6) and wastewater (7) or for water treatment optimization (8). PM discretization as a function of heterodispersivity is important in modeling treatment and fate of PM and PM-bound pollutants. A common analysis for PM separation by treatment unit operations is the overflow rate theory with the inherent assumption of type I gravitational settling, where PM settles
discretely with negligible particle-particle interaction and impact on the fluid flow field. This basic theory can be combined with a constitutive model of PM settling described by Newton’s law. Coupling overflow rate, Newton’s Law and PSD discretization, while less common, permits a more complete description of discrete PM settling irrespective of the continuous fluid phase dynamics. The most basic PSD discretization and separation representation is using a massbased median size (d50m), with increasingly accurate representation generated by higher PSD discretization. While the coupling of a representative PSD and Newton’s Law with overflow rate provides a more accurate description of the discrete PM phase, a major shortcoming of the basic overflow rate concept is the lack of a quantitative coupling with the potential hydrodynamic complexity of the fluid phase. Coupling the dynamics of the fluid phase can range from semiempirical overflow rate models such as Hazen’s law (9, 10) to a fundamental description with Navier-Stokes equations (11). Computational fluid dynamics (CFD) has significantly improved modeling of PM separation and fate for complex geometries, nonideal flow fields, and transient flows (12, 13). However, the level of PM discretization as a function of PSD heterodispersivity is required when modeling unit operations with a Lagrangian-Eulerian CFD approach. PM laden flows represent a challenging modeling phenomenon with discrete PM sizes, as PSDs can be considered a continuum. Using a discrete phase model (DPM) to simulate PM separation requires discrete PM sizes for a continuous PSD. As heterodispersivity increases, the particle number exponentially increases with increasing computational effort. Identifying PSD discretization needed for acceptable results and computational economy is important.
Objectives A primary study objective is the examination of PM discretization requirements in a CFD model for selected levels of granulometric size heterodispersivity. Additionally, this study illustrates the impact of gradation uniformity and overall gradation fineness on PSD discretization requirements for CFD modeling of two different hydrodynamic separators (HS) commonly utilized worldwide for treatment of urban drainage. Using controlled physical modeling of a baffled HS and screened HS shown in Figure 1, to validate the CFD model, the study hypothesized that the error in modeling PM separation is a function of the PSD discretization.
Materials and Methods A granulometric attribute of PM is the PSD. Urban drainage PM in wet weather (rainfall-runoff) or dry weather (wastewater) flows is heterodisperse. To explore the effect of PSD dispersivity on PSD discretization requirements a rubric is needed to characterize the gradation uniformity. Folk and Ward (14) proposed a sorting coefficient (σI) for this granulometric attribute. With the sorting coefficient, gradations of similar uniformity can be generated at a chosen d50m. Equation 1 presents the sorting coefficient modified by a negative sign since percentiles are reported as % finer by mass whereas Folk and Ward present them as % greater. σI ) -
* Corresponding author e-mail:
[email protected]. 8220
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φ84 - φ16 φ95 - φ5 + 4 6.6
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2009 American Chemical Society
Published on Web 10/02/2009
FIGURE 1. Experimental validation of the change in event mean concentration for the screened HS (a) and baffled HS (b) CFD models from experiments conducted on full-scale 1.8 m (6 ft) diameter units under laboratory conditions, loaded with the heterodisperse NJ Department of Environmental Protection (NJDEP) regulatory gradation (gradation no. 8). As required, the CFD model results were linearly interpolated to provide data points at concurrent flow rates for RPD calculation. Range bars show experimental mass balance recovery (%). Influent flow is demarcated by Q in each subfigure. Reported volumes are calculated from the water level under static conditions. In this expression φ is the phi-scale particle size gradation parameter (15), and is defined by φn ) -log2
dn d0
(2)
Where n is the percentile, φn is the phi parameter of the nth finer percentile, dn is the PM diameter of the nth finer percentile, and d0 is the unit length to nondimensionalize the equation. In order to systematically study the effect of PSD discretization for size gradations of differing σI and d50m, a methodology is needed to generate PSDs that vary only in these two parameters while simultaneously generating representative urban drainage PM gradations. Previous studies (16, 17) have utilized a two parameter cumulative gamma distribution to model heterodisperse urban drainage PSDs. The shape and scale factor parameters in the cumulative gamma distribution, k and λ, are physically analogous to σI and d50m. Equations 3 and 4 represent the gamma distribution (f(x)) and cumulative gamma distribution (F(x)), respectively. x ( λ) f (x) )
k-1 (-x/λ)
e
λ · Γ(k)
F(x) )
∫ f (x)dx x
0
(3)
(4)
Where k is the gamma shape factor; λ is the gamma scale factor; and x is the particle diameter.
Selected Particle Size Distributions. Table 1 and Figure 2 describe the chosen gradations for this study. Gradation characteristics are selected to elucidate the effects of PSD dispersivity and also the PSD d50m on the CFD simulation results for different levels of PSD discretization using a discretization number (DN) at a fixed density (F ) 2.65 g/cm3; silica sand). The nine gradations are generated by identifying gamma distribution k and λ parameters that matched the 3 × 3 gradation matrix of monodisperse (σI < 0.35; in this study: σI ) 0.11), moderately disperse (σI ≈ 1.00; in this study: σI ) 1.03) and heterodisperse (σI > 2.00; in this study: σI ) 2.64) PSDs. The gravimetric median sizes for these PSDs are coarse (d50m ) 100 µm), fine (d50m ) 66.7 µm), and very fine (d50m ) 33.3 µm). Computational Fluid Dynamics (CFD). CFD simulates behavior of unit operations at a variety of scales, from pilot-scale behavior to smaller-scale phenomena, such as the internal flow field velocities and pressure distributions as well as PM transport and fate. CFD is utilized to model behavior of each HS as a function of granulometric attributes (PSD, d50m and σI) and flow. CFD is based on numerical solutions to Navier-Stokes equations across a domain. Specifically, to model the flow fields in this study, Reynolds Averaged Navier-Stokes (RANS) equations (18) are utilized with Fluent 6.3.26. The RANS equations decompose the bulk, time independent fluid flow from the transient turbulent fluctuations. Averaged over a timescale much larger than the time-scale of the fluctuations, the transient turbulent fluctuations become zero leaving only the bulk fluid flow and stationary turbulence structures. Equations 5 and 6 below, succinctly describe the VOL. 43, NO. 21, 2009 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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TABLE 1. Characteristics of Selected Gradations Including Gamma Distribution Parameters (k, λ), and the Sorting Coefficient (σI)a table of cumulative gamma distribution modeled gradations uniform gradation no. k λ σI F (g/cm3) PSD Indice d95m d84m d50m d16m d5m
medium
1
2
3
162.94 0.20 0.11 2.65
177.19 0.38 0.11 2.65
163.07 0.61 0.11 2.65
2.28 17.01 1.03 2.65
4
37.8 35.9 33.3 30.8 29.2
75.3 71.8 66.7 61.8 58.8
113.4 107.9 99.9 92.3 87.6
88.4 62.5 33.3 15.1 8.1
5
heterodisperse 6
2.30 2.30 33.82 50.33 1.03 1.03 2.65 2.65 Particle Size (µm) 176.8 262.8 125.2 186.1 67.0 99.4 30.6 45.4 16.3 24.2
7
8
9
0.57 116.32 2.63 2.65
0.56 232.64 2.64 2.65
0.56 353.16 2.65 2.65
241.9 128.9 33.2 3.8 0.5
483.2 257.3 66.1 7.5 0.9
730.6 388.4 99.3 11.2 1.4
a Gradation no. 8 is the gamma curve fit of the NJDEP gradation (R2 ) 0.99). The other gradations were selected to systematically explore the effect of uniformity and d50m on discretization error and were chosen so that the uniform gradations are very well sorted (σI < 0.23), the medium gradations transect the boundary between moderately sorted and poorly sorted gradations (σI ) 1.0), and the other hetero-disperse gradations (no. 7, no. 9) have similar sorting to the NJDEP distribution (no. 8). The distribution indices (percentiles) necessary for the calculation of the sorting coefficient are also included. The power law model coefficients for both the mean and maximum RPD models are summarized in Supporting Information Table S3. The PSDs are presented in Figure 2.
steady state RANS continuity and momentum equations respectively:
F
∂ (Fu j i) ) 0 ∂xi
(5)
∂2u ji ∂ ∂p j ∂ (uiuj) + F (u′ju′i) ) + µ 2 + Fgi ∂xj ∂xj ∂xi ∂xj
(6)
In these equations F is fluid density; xi is the ith direction vector; uji is the Reynolds averaged velocity in the ith direction; pj is the Reynolds averaged pressure; and gi is the sum of body forces in the ith direction (i.e., gravity in the negative z direction). The decomposition of the nonlinear convection in the momentum equation results in Reynolds stresses - Fu′ju′l. The Reynolds stresses are unknown quantities and can be modeled with the semiempirical k - ε model, presented below, that was developed by Shih et al. (19) which performs well for rotating homogeneous (screened HS) and boundary-free (baffled HS) shear flows:
( ) ( )
∂u ji ∂ vT ∂k ∂k ∂k ) - u′iu′j -ε +u jj ∂t dxj ∂xj σk ∂xj ∂xj
(7)
∂ vT ∂ε ∂ε ε2 ∂ε ) + C1Sε - C2 +u jj ∂t dxj ∂xj σε ∂xj k + √vε
(8)
Discrete Phase Model (DPM). To model transport of PM within HS units, a mixed mode Eulerian-Lagrangian reference frame is utilized where fluid velocity and pressure flow fields are modeled in an Eulerian or control volume reference frame and PM is modeled as discrete particles in a Lagrangian or particle tracking reference frame. PM transport modeled in the discrete phase is integrated across the fluid velocity and pressure flow fields modeled in the Eulerian reference frame. This does not account for particle influence on velocity and pressure flow fields and is restricted to dilute fluid flows with
