Solute Mixing Models for Water-Distribution Pipe Networks

The spreading of solutes or contaminants through water-distribution pipe networks is controlled largely by mixing at pipe junctions where varying flow rates and concentrations can enter the junction. Alternative models of solute mixing within these pipe junctions are presented in this paper. Simple complete-mixing models are discussed along with rigorous computational-fluid-dynamics models based on turbulent Navier–Stokes equations. In addition, a new model that describes the bulk-mixing behavior resulting from different flow rates entering and leaving the junction is developed in this paper. Comparisons with experimental data have confirmed that this bulk-mixing model provides a lower bound to the amount of mixing that can occur within a pipe junction, while the complete-mixing model yields an upper bound. In addition, a simple scaling parameter is used to estimate the actual (intermediate) mixing behavior based on the bounding predictions of the complete-mixing and bulk-mixing models. These simple analytical models can be readily implemented into network-scale models to develop predictions and bounding scenarios of solute transport and water quality in water-distribution systems.     

Eulerian–Lagrangian Method for Constituent Transport in Water Distribution Networks

The transport and mixing of contaminants in conduits is governed by advection, dispersion, and decay. Several models are available to trace the transport of such constituents and most assume that the principal mechanisms for transport are advection and reaction only. However in pipes where low velocities prevail, longitudinal dispersion is significant and models that neglect the dispersion effects fail to properly simulate the observed concentrations in low velocity pipes. This work presents a method for simulating the advection-dispersion-reaction process of constituent transport in water networks. A Eulerian–Lagrangian method is employed whereby the dispersion term in the governing equation is approximated using finite differences and the resulting first-order partial differential equation is then integrated using the method of characteristics. Analytical solutions of the transport equation are also derived to quantify the effect of neglecting dispersion at pipe junctions and to assess the accuracy of the proposed method. The Eulerian-Lagrangian method is tested on benchmark networks and on the field study at the Cherry Hill/Brushy Plains network. Results show that the model developed is capable of simulating transport with equal accuracy for low and high velocity flows with and without significant dispersion effects. It also performs better than other models because of the nonuniform grid distribution and the interpolation schemes used.     

Pollutant Transport and Mixing Zone Simulation of Sediment Density Currents

Prediction of water column concentrations of suspended sediment is often necessary for environmental impact assessment of point source industrial discharges. For example, in “flow lane” or “open water” disposal, suction dredges discharge large volumes of suspended sediment into shallow water disposal locations. A sediment density current mixing model is presented here as part of the D-CORMIX expert system for hydrodynamic simulation of mixing zone behavior. This density current model extends the CORMIX decision support system to simulate continuous negatively buoyant discharges with or without suspended sediment loads on a sloping bottom with loss of suspended particles by sedimentation. Sedimentation is modeled using Stokes settling for five particle size classes. Density current width and depth, trajectory, total solids, tracer concentration, dilution, and particle size concentration are predicted. In addition, location and widths of sediment deposits, accretion rates, including particle size fractions within the spoils deposit, are predicted. The model results are in good overall agreement with available field and laboratory data.     

Dispersion Model for Tidal Wetlands

Tidal wetlands in California are mostly estuarine salt marshes characterized by tidal channels and mudflats that are flooded and drained on a semidiurnal basis. Depths are rarely greater than 2 or 3 m, except where dredging occurs for harbor operations, and lengths from head to mouth are usually in the range of 1–10 km. This paper presents a coupled set of models for prediction of flow, solute transport, and particle transport in these systems. The flow and solute transport models are based upon depth-integrated conservation equations while the particle transport model is quasi-three-dimensional. Common to these models is an assumption that a turbulent boundary layer extends vertically from the bed and can be described by the law of the wall. This feature of the model accounts for: (1) momentum transfer to the bed, (2) longitudinal dispersion of dissolved material based on the work of Elder (1959), and (3) advection and turbulent diffusion of particles in three dimensions. A total variation diminishing finite volume scheme is used to solve the depth-integrated equations. Using this model, we show that dispersion can be accurately modeled using physically meaningful mixing coefficients. Calibration is therefore directed at modifying bed roughness, which scales both the rate of advection and dispersion.     

Evaluation of Mixing and Performance of Lab-Scale Upflow Anaerobic Sludge Blanket Reactors Treating Domestic Wastewater

The effects of varying hydraulic retention time (HRT) and associated upflow velocity on mixing and reactor performance were evaluated in five lab-scale upflow anaerobic sludge blanket (UASB) reactors treating real domestic wastewater. The mixing and transport studies were carried out with the help of tracer experiments at various HRTs using a pulse tracer input. A number of existing models were assessed for the analysis of the time series of observed tracer concentrations. The plug-flow reactor (PFR) model with two-zone dispersion better simulated the time series of tracer concentrations at all HRTs than other models, such as single compartment dispersion, completely mixed flow reactors (CMFRs) in series, and a combination of CMFR and PFR. The dispersion coefficients obtained from the two-zone dispersion model correlated well with the dispersion analysis expression for flow in a circular cylinder, and the correlation can be used for the prediction of dispersion in a UASB reactor. The analysis of reactor performance data indicated that reduction of dispersion owing to decrease in the upflow velocity resulted in increased sulfidogenic activity in the reactor. This was attributed to the inability of the sulfate reducers to colonize in the reactor at high upflow velocity and mixing condition.     

Goodness-of-Fit Test for Modeling Tracer Breakthrough Curves in Wetlands

Wetland transport models generally either assume plug flow (with or without dispersion) or conceptualize the wetland as a series of continuously stirred tank reactors (CSTRs). To evaluate the CSTR approach, we present a goodness-of-fit test suitable for evaluating breakthrough curves from tracer experiments. The test, which makes use of confidence intervals associated with the multivariate normal distribution, can be used to test the fit of the breakthrough curve model, but requires sampling across a transect rather than from a single point. To test the CSTR assumption, we conducted a pair of two-dimensional tracer experiments within a 9.9?ha wetland constructed to receive effluent from a wastewater treatment plant in San Jacinto, Calif. The wetland operates with five parabolic inlets and a single large parabolic outlet to encourage lateral uniformity. In both experiments tritium oxide (HTO) was used as the tracer. Rhodamine WT dye was also included in the second experiment. Tracer samples were collected along transects installed perpendicular to the direction of flow. Analysis of the results indicates satisfactory lateral mixing and no significant short-circuiting. Rhodamine WT dye performed similarly to HTO when detectable but was too dilute to be observed at the outlet. When tracer movement was modeled as a series of continuously stirred reaction vessels, the parameter associated with the integer number of vessels increased from 2 at the first transect to 8 at the outlet. At each transect, the model was checked with a new goodness-of-fit test. At the α = 0.05 confidence level, all fitted models were rejected, suggesting that while the CSTR assumption may usefully approximate transport processes, it is not statistically valid for this wetland.     

Predictive Modeling of Transient Storage and Nutrient Uptake: Implications for Stream Restoration

This study examined two key aspects of reactive transport modeling for stream restoration purposes: the accuracy of the nutrient spiraling and transient storage models for quantifying reach-scale nutrient uptake, and the ability to quantify transport parameters using measurements and scaling techniques in order to improve upon traditional conservative tracer fitting methods. Nitrate (NO3?) uptake rates inferred using the nutrient spiraling model underestimated the total NO3? mass loss by 82%, which was attributed to the exclusion of dispersion and transient storage. The transient storage model was more accurate with respect to the NO3? mass loss (±20%) and also demonstrated that uptake in the main channel was more significant than in storage zones. Conservative tracer fitting was unable to produce transport parameter estimates for a riffle-pool transition of the study reach, while forward modeling of solute transport using measured/scaled transport parameters matched conservative tracer breakthrough curves for all reaches. Additionally, solute exchange between the main channel and embayment surface storage zones was quantified using first-order theory. These results demonstrate that it is vital to account for transient storage in quantifying nutrient uptake, and the continued development of measurement/scaling techniques is needed for reactive transport modeling of streams with complex hydraulic and geomorphic conditions.     

Effectiveness of Different Artificial Neural Network Training Algorithms in Predicting Protozoa Risks in Surface Waters

A neural network approach was employed to relate risky Cryptosporidium and Giardia concentrations with other biological, chemical and physical parameters in surface water. A set of drinking water samples was classified as “risky” and “nonrisky” based on the concentrations of full and empty oocysts, and cycsts of Cryptosporidium and Giardia, respectively. Given the constraints associated with collecting large sets of microbial data, the study was aimed at identifying an effective training algorithm that would maximize the performance of a neural network model working with a relatively small dataset. A number of algorithms for training neural networks, including gradient search with first- and second-order partial derivatives, and genetic search were used and compared. Results showed that genetic algorithm based neural network training consistently provided better results compared to other training methods.     

Simplified Numerical and Analytical Approach for Solutes in Turbulent Flow Reacting with Smooth Pipe Walls

A large group of reactions that affect water quality in distribution networks occur on the pipe wall surface. Existing simulation models are usually based on cross-sectionally averaged variables that use mass-transfer coefficients derived for constant-concentration (Dirichlet) boundary conditions to account for cross-sectional variations. In the case of a first-order wall-demand problem, the boundary condition is however of Robin type. We derive a simple one-dimensional (1D) model for the radial concentration profile of a solute of arbitrary Schmidt number (Sc) reacting with pipe walls in a fully developed turbulent flow. A modified van Driest mixing length model was used to approximate the Reynolds-averaged velocity and eddy diffusivity. Numerical solutions of the 1D model agree well with a two-dimensional mass transport model and experimental data. An asymptotic solution for high Sc is derived, which is in excellent agreement with the 1D model for Sc>100. A comparison with the mass-transfer coefficients for constant-concentration boundary conditions shows that the differences between the two boundary conditions are small.     

Influent Pollutant Concentrations as Predictors of Effluent Pollutant Concentrations for Mid-Atlantic Bioretention

The water quality performance of best management practices (BMPs) has been frequently assessed by the removal efficiency metric. Recent findings show that the removal efficiency metric is flawed because it does not account for background water quality, eco-region differentiation, and background, or “irreducible,” concentrations. Additionally, the removal efficiency metric inherently assumes a definite association exists between influent and effluent pollutant concentrations. Such a relationship between influent and effluent concentrations has been minimally studied for bioretention, the most common storm-water control measure associated with low-impact development (LID). This study analyzes influent and effluent total nitrogen (TN) and total phosphorous (TP) concentrations from 11 bioretention cells in the mid-Atlantic United States. Pooled data showed only a slight association between influent and effluent TN. Essentially no relationship exists between influent and effluent TP concentration. Both findings indicate that the percent-removal metric is a faulty means of evaluating bioretention performance. Twelve general linear models (GLMs) were created where influent TN and TP were the predictors of respective effluent TN and TP concentrations. Only one GLM was considered to be “good,” defined as 67–90% of the variation in effluent concentrations being explained by respective influent concentrations (R2 = 0.72). In addition, there were two “fair” models, five “poor” models, and four “very poor” models. No “very good” models were found for TN or TP. Furthermore, as influent nutrient concentration in runoff increases, the removal efficiency increases for TN and TP. “Dirtier” influent TP concentrations were effectively reduced; conversely, “cleaner” TP influent concentrations increased, both tending toward a (possibly media-controlled) baseline effluent concentration (0.10 to 0.18??mg/l). TN effluent data also may have been tending toward a common concentration; however, the value was not as discernible.     

Velocity Pulse Model for Turbulent Diffusion from Flowing Water into a Sediment Bed

The “velocity pulse model” simulates the transfer of turbulence from flowing water into a sediment bed, and its effect on the diffusional mass transfer of a solute (e.g., oxygen, sulfate, or nitrate) in the sediment bed. In the “pulse model,” turbulence above the sediment surface is described by sinusoidal variations of vertical velocity in time. It is shown that vertical velocity components dampen quickly inside the sediment when the frequency of velocity fluctuations is high and viscous dissipation is strong. Viscous dissipation (ν) inside the sediment is related to the apparent viscosity depending on the structure of the sediment pore space, i.e., the porosity and grain diameter, as well as inertial effects when the flow is turbulent. A value ν/ν0 between 1 and 20 (ν0 is kinematic viscosity of water) has been considered. Turbulence penetration into the sediment is parametrized by the Reynolds number Re = UL/ν and the relative penetration velocity W/U, where U=amplitude of the velocity pulse; and W=penetration velocity; L = WT=wave length of the velocity pulse; and T is its period. Amplitudes of vertical velocity components inside the sediment and their autocorrelation functions are computed, and the results are used to estimate eddy viscosity inside the sediment pore system as a function of depth. Diffusivity in the sediment pore system is inferred by using turbulent or molecular Schmidt numbers. Turbulence penetration from flowing water can enhance the vertical diffusion coefficient in a sediment bed by an order of magnitude or more. Penetration depth of turbulence is higher for low frequency velocity pulses. Vertical diffusivity inside the pore system is shown to decrease more or less exponentially with depth below the sediment/water interface. Vertical diffusivities in a sediment bed estimated by the “velocity pulse model” can be used in pore water quality models to describe vertical transport from or into flowing surface water. The analysis has been conducted for a conservative material, but source and sink terms can be added to the vertical transport equation.     

Analysis and Prediction of Transverse Mixing Coefficients in Natural Channels

An experimental study has been undertaken using a naturally formed meandering channel to obtain unique tracer and hydrodynamic data. The velocity data presented are from laser Doppler anemometer measurements; tracer data were collected using an array of fluorometers in continuous flow through mode. Techniques for the prediction of the primary and secondary velocity flow fields are explored, and shown to be accurate. Analysis of the tracer data by the generalized method of moments using the cumulative transverse discharge approach is undertaken. The coefficient of transverse mixing is shown to exhibit considerable longitudinal variation over the meander cycle. A new integrated approach for predicting transverse mixing coefficients is developed and explored and has been validated against the data set. This approach requires only three parameters as input, namely, longitudinal planform curvature, cross-sectional shape, and total discharge, and has been shown capable of accurately predicting the longitudinal variation of the transverse mixing coefficient over the meander cycle.     

Case Study: Modeling Tidal Transport of Urban Runoff in Channels Using the Finite-Volume Method

A coupled flow and pollutant transport model based on the finite-volume method is developed and applied to predict the tidal transport of urban runoff in a southern California network of flood control channels that drain to near-shore bathing waters. Urban runoff in southern California contains elevated levels of indicator bacteria that signal the presence of fecal pollution and pose a risk to human health, and model predictions are used to understand the transport of these pollutants toward the coastline. The model is based on 1D conservation equations for fluid mass, momentum, and pollutant mass that are solved in integral form along channel reaches. A 2D formulation is solved at channel junctions. The model incorporates the monotone upwind scheme for conservation laws approach to give a high-resolution, nonoscillatory prediction of water level, velocity, and pollutant concentration. Model predictions and field measurements of water level, velocity, and a conservative urban runoff tracer are presented and compare favorably. This case study demonstrates that this finite-volume method–based scheme results in an accurate, stable, nonoscillatory and computationally manageable model. The nonoscillatory behavior is particularly beneficial in this study, since runoff enters the channels in pulses that create large gradients in pollutant concentration.     

THE DENSITY OF PACKING OF TWO-COMPONENT POWDER MIXTURES

Abstract

The problem of the bulk density attained by a mixture of two powders, of uniform particle size, is considered. It is shown that there are three limiting cases for which the bulk density may easily be calculated and that, from these cases, the bulk density corresponding to a wide range of conditions may be deduced.

It is suggested that a clear distinction must be maintained between a perfect ordered mix, designated a “hyperperfect” mix, and a perfect randomized mix, the randomized mix being the only type attainable by use of a practical mixing machine. The perfection of the mixing of the two component powders is an important factor in the problem and, for mixes as obtained from practical mixing machines, the bulk density is considerably lower than would be the case with the theoretically perfect randomized mix.

The assumption upon which most of the work on the density of mixtures is based, namely that of a “hyperperfect mix”, is inappropriate to industrial mixing processes, so that such treatments are invalid.     

Optimal Design of Pressurized Irrigation Subunit

A linear programming (LP) model is presented for optimal design of the pressurized irrigation system subunit. The objective function of the LP is to minimize the equivalent annual fixed cost of pipe network of the irrigation system and its annual operating energy cost. The hydraulic characteristics in the irrigation subunit are ensured by using the length, energy conservation, and pressure head constraints. The input data are the system layout, segment-wise cost and hydraulic gradients in all the alternative pipe diameters, and energy cost per unit head of pumping water through the pipeline network. The output data are: segment-wise lengths of different diameters, operating inlet pressure head, and equivalent annual cost of the pipeline network. The explicit optimal design is demonstrated with design examples on lateral and submain or manifold of pressurized irrigation systems. The effect of the equations for friction head loss calculation on optimization procedure is investigated through the design example for microirrigation manifold. The performance evaluation of the proposed model in comparison with the analytical methods, graphical methods, numerical solutions, and dynamic programming optimization model reveals the good performance of the proposed model. The verification of operating inlet pressure head obtained by the proposed model with accurate numerical step-by-step method suggested that it is mostly accurate.     

One-Dimensional Mixing Model for Surcharged Manholes

Mixing and dispersion processes affect the timing and concentration of contaminants transported within urban drainage systems. Hence, methods of characterizing the mixing effects of specific hydraulic structures are of interest to drainage network modelers. Previous research, focusing on surcharged manholes, used the first-order advection-dispersion equation (ADE) and aggregated dead zone (ADZ) models to characterize dispersion. However, although systematic variations in travel time as a function of discharge and surcharge depth were identified, the ADE and ADZ models did not provide particularly good fits to observed manhole mixing data, which meant that the derived parameter values were not independent of the upstream temporal concentration profile, and no rules for predicting parameter values based on manhole size and configuration were provided. An alternative, more robust, method is described by using the system’s cumulative residence time distribution (CRTD). This paper shows how a deconvolution approach derived from systems theory may be applied to identify, from laboratory data, the CRTDs associated with surcharged manholes. Archive laboratory data are reanalyzed to demonstrate that the solute transport characteristics of a surcharged manhole with straight-through inflow and outlet pipes over a range of flow rates and surcharge depths may be modeled using just two dimensionless CRTDs, one for prethreshold and the other for postthreshold surcharge depths. The model combines the derived manhole CRTDs with a standard (Gaussian) pipe dispersion model to provide temporal solute concentration profiles that are independent of both scale and the ratio of the pipe and manhole diameters.     

Modeling Residual Chlorine Response to a Microbial Contamination Event in Drinking Water Distribution Systems

Changes in chlorine residual concentrations in water distribution systems could be used as an indicator of microbial contamination. Consideration is given on how to model the behavior of chlorine within the distribution system following a microbial contamination event. Existing multispecies models require knowledge of specific reaction kinetics that are unlikely to be known. A method to parameterize a rate expression describing microbially induced chlorine decay over a wide range of conditions based on a limited number of batch experiments is described. This method is integrated into EPANET-MSX using the programmer’s toolkit. The model was used to simulate a series of microbial contamination events in a small community distribution system. Results of these simulations showed that changes in chlorine induced by microbial contaminants can be observed throughout a network at nodes downstream from and distant to the contaminated node. Some factors that promote or inhibit the transport of these chlorine demand signals are species-specific reaction kinetics, the chlorine concentration at the time and location of contamination, and the system’s unique demand patterns and architecture.     

Aldicarb Transport in Subsurface Environment: Comparison of Models

In this paper, two different pesticide transport simulation models are presented and compared to carry out preliminary analysis on the applicability of those models in determining ground-water vulnerability to aldicarb contamination. The first model is a physically based analytical model that simulates 1D pesticide movement in soils, based on the concept of complete mixing and 2D advective-dispersive transport in the aquifer. The second model is a numerical simulation model that links the existing numerical codes PRZM2, MODFLOW, and MT3D to simulate pesticide transport in the subsurface. The concentrations of aldicarb residues in soil and in the aquifer calculated by the two models are found to be in good agreement. However, the analytical model tends to produce an earlier arrival of the peak concentration in each year due to the assumption of complete mixing. It is also found that the infiltrating water following aldicarb application plays a significant role on the leaching potential of aldicarb, which is also affected by various meteorological and hydrological factors as well as by agricultural practices.     

Probabilistic Behavior of Water-Quality Loads

A theoretical model is introduced for describing the mechanistic and probabilistic structure of observations of streamflow Q, concentration C, and constituent loads L. The model has application to many water-quality management problems including load estimation, water-quality monitoring network design and total maximum daily load assessment. The statistical behavior of streamflow, concentration, and load is described and expressions are derived for the coefficient of variation of daily concentrations and loads assuming a bivariate lognormal model. The model provides a first-order approximation to continuous empirical observations of C, Q, and L from four watersheds in the Great Lakes Region. The utility of the model is demonstrated by quantifying the amount of “spurious” correlation between load and discharge, by documenting factors which influence bias in water-quality load estimates and those which give rise to increased/decreased variability in water-quality loads and concentrations.