Solving Mass Balance Equations Research Articles

A multimedia model to estimate the environmental fate of microplastic particles

Nano- and microplastic (NMP) is a diverse and challenging contaminant and data on NMP concentrations are therefore not fully available for all environmental compartments. For environmental assessments of NMP, screening-level multimedia models can fill this gap, but such models are not available. Here, we present SimpleBox4Plastic (SB4P) as the first multimedia ‘unit world’ model capable of addressing the full NMP continuum, explore its validity, and evaluate it based on a case study for microbeads and by comparisons with (limited) concentration data. SB4P links NMP transport and concentrations in and across air, surface water, sediment, and soil, taking into account processes such as attachment, aggregation, and fragmentation, by solving mass balance equations using matrix algebra. These link all concentrations and processes known to be relevant for NMP using first-order rate constants, which are obtained from the literature. The SB4P model, as applied to microbeads, provided mass or number concentrations of NMP as the total of ‘free’ particles, heteroaggregates with natural colloids, and larger natural particles in each compartment at steady state. Processes most relevant in explaining observed Predicted Exposure Concentrations (PECs) were determined using rank correlation analysis. Although the predicted PECs remained uncertain due to the propagating uncertainty, inferences regarding these processes and relative distribution across compartments can be considered robust.

Experimental and modelling of Arthrospira platensis cultivation in open raceway ponds

In this study, the growth of Arthrospira platensis was studied in an open raceway pond. Furthermore, dynamic model for algae growth and CFD modelling of hydrodynamics in open raceway pond were developed. The dynamic behaviour of the algal system was developed by solving mass balance equations of various components, considering light intensity and gas-liquid mass transfer. A CFD modelling of the hydrodynamics of open raceway pond was developed by solving mass and momentum balance equations of the liquid medium. The prediction of algae concentration from the dynamic model was compared with the experimental data. The hydrodynamic behaviour of the open raceway pond was compared with the literature data for model validation. The model predictions match the experimental findings. Furthermore, the hydrodynamic behaviour and residence time distribution in our small raceway pond were predicted. These models can serve as a tool to assess the pond performance criteria.

Dynamic modelling of microalgae cultivation process in high rate algal wastewater pond

In this work, a comprehensive dynamic mathematical modelling to simulate the production of microalgae in a high rate algal pond (HRAP) is attempted. A synergetic algal–bacterial system comprising various interrelated biological and chemical system processes is presented. The dynamic behaviour of HRAP system is studied by solving mass balance equations of different components which account light intensity and gas–liquid mass transfer. The model predictions are compared with the previously reported studies in the literature. The influence of kinetic and operating parameters, including the supply of CO2, the maximum growth rate, pond depth and dilution rates, on the pond performance are evaluated. The sensitivity analysis of important process parameters is also discussed in this study. The developed model, as a tool, can be used to assess the factors that affect the pond performance criteria, including algal productivity and the dynamics of nutrient requirements.

Methodological uncertainty in resource mixing models for generalist fishes

Carbon and nitrogen stable isotope ratios are used to assess diet composition by determining bounds for the relative contributions of different prey to a predator's diet. This approach is predicated on the assumption that the isotope ratios of predator tissues are similar to those of dominant food sources after accounting for trophic discrimination (Δ(x)X), and is formulated as linear mixing models based on mass balance equations. However, Δ(x)X is species- and tissue-specific and may be affected by factors such as diet quality and quantity. From the different methods proposed to solve mass balance equations, some assume Δ(x)X to be exact values whilst others (based on Bayesian statistics) incorporate variability and inherent uncertainty. Using field data from omnivorous reef fishes, our study illustrates how uncertainty may be taken into account in non-Bayesian models. We also illustrate how dietary interpretation is a function of both absolute Δ(x)X and its associated uncertainty in both Bayesian and non-Bayesian isotope mixing models. Finally, collated literature illustrate that uncertainty surrounding Δ(x)X is often too restricted. Together, these data suggest the high sensitivity of mixing models to variation in trophic discrimination is a consequence of inappropriately constrained uncertainty against highly variable Δ(x)X. This study thus provides guidance on the interpretation of existing published mixing model results and in robust analysis of new resource mixing scenarios.

Post-breakthrough scaling in reservoir field simulation

We study the oil displacement and production behavior in an isothermal thin layered reservoir model subjected to water flooding. We use the CMG’s (Computer Modelling Group) numerical simulators to solve mass balance equations. The influences of the viscosity ratio (m≡μoil/μwater) and the inter-well (injector-producer) distance r on the oil production rate C(t) and the breakthrough time tbr are investigated. Two types of reservoir configuration are used, namely one with random porosities and another with a percolation cluster structure. We observe that the breakthrough time follows a power-law of m and r, tbr∝rαmβ, with α=1.8 and β=−0.25 for the random porosity type, and α=1.0 and β=−0.2 for the percolation cluster type. Moreover, our results indicate that the oil production rate is a power law of time. In the percolation cluster type of reservoir, we observe that P(t)∝tγ, with γ=−1.81, where P(t) is the time derivative of C(t). The curves related to different values of m and r may be collapsed suggesting a universal behavior for the oil production rate.

Comparison of SOC estimates and uncertainties from aerosol chemical composition and gas phase data in Atlanta

In the Southeastern US, organic carbon (OC) comprises about 30% of the PM 2.5 mass. A large fraction of OC is estimated to be of secondary origin. Long-term estimates of SOC and uncertainties are necessary in the evaluation of air quality policy effectiveness and epidemiologic studies. Four methods to estimate secondary organic carbon (SOC) and respective uncertainties are compared utilizing PM 2.5 chemical composition and gas phase data available in Atlanta from 1999 to 2007. The elemental carbon (EC) tracer and the regression methods, which rely on the use of tracer species of primary and secondary OC formation, provided intermediate estimates of SOC as 30% of OC. The other two methods, chemical mass balance (CMB) and positive matrix factorization (PMF) solve mass balance equations to estimate primary and secondary fractions based on source profiles and statistically-derived common factors, respectively. CMB had the highest estimate of SOC (46% of OC) while PMF led to the lowest (26% of OC). The comparison of SOC uncertainties, estimated based on propagation of errors, led to the regression method having the lowest uncertainty among the four methods. We compared the estimates with the water soluble fraction of the OC, which has been suggested as a surrogate of SOC when biomass burning is negligible, and found a similar trend with SOC estimates from the regression method. The regression method also showed the strongest correlation with daily SOC estimates from CMB using molecular markers. The regression method shows advantages over the other methods in the calculation of a long-term series of SOC estimates.

A GIS-based Approach for Modeling the Fate and Transport of Pollutants in Europe

This paper presents an approach to estimate chemical concentration in multiple environmental media (soil, water, and the atmosphere) with the sole use of basic geographical information system (GIS) operations and, particularly, map algebra. This allows solving mass balance equations in a different way from the traditional methods involving numerical or analytical solution of systems of equations, producing maps of chemical fluxes and concentrations only through combinations of maps of emissions and environmental removal or transfer rates. Benchmarking with the well-established EMEP MSCE-POP model shows that the method provides consistent results with this more detailed description. When available, experimental evidence equally supports the proposed method in relation to the more complex approaches.

Development of a Multichemical Food Web Model: Application to PBDEs in Lake Ellasjøen, Bear Island, Norway

A multichemical food web model has been developed to estimate the biomagnification of interconverting chemicals in aquatic food webs. We extended a fugacity-based food web model for single chemicals to account for reversible and irreversible biotransformation among a parent chemical and transformation products, by simultaneously solving mass balance equations of the chemicals using a matrix solution. The model can be applied to any number of chemicals and organisms or taxonomic groups in a food web. The model was illustratively applied to four PBDE congeners, BDE-47, -99, -100, and -153, in the food web of Lake Ellasjøen, Bear Island, Norway. In Ellasjøen arctic char (Salvelinus alpinus), the multichemical model estimated PBDE biotransformation from higher to lower brominated congeners and improved the correspondence between estimated and measured concentrations in comparison to estimates from the single-chemical food web model. The underestimation of BDE-47, even after considering bioformation due to biotransformation of the otherthree congeners, suggests its formation from additional biotransformation pathways not considered in this application. The model estimates approximate values for congener-specific biotransformation half-lives of 5.7,0.8,1.14, and 0.45 years for BDE-47, -99, -100, and -153, respectively, in large arctic char (S. alpinus) of Lake Ellasjøen.

A computer program for analysing complex bulk XAFS spectra and for performing significance tests.

LSFitXAFS is a computer program designed to calculate fractions of individual components (i.e. individual species, minerals, compounds or chemical environments) making up bulk XAFS spectra. The program performs Gaussian elimination and multiple linear regression techniques to simultaneously solve mass balance equations. In addition, the program contains a number of data reduction and analysis routines including the determination of random and systematic noise in Fourier-transformed spectra and calculation of the goodness-of-fit parameters for different combinations of shells.

Solution equilibria of deferoxamine amides

The physico‐chemical solution properties of deferoxamine were modified by acylating the terminal amino group with short‐chain aliphatic, succinic, and methylsulphonic moieties. The analog iron(III)‐binding constants and stabilities under physiological conditions were determined to confirm that the iron binding ability of the parent molecule was retained following modification. The proton dissociation constants of the lipophilic deferoxamine analogs were determined by potentiometric titration and nonlinear least‐squares analysis. However, because the iron(III) binding complex is fully formed below pH 2, the metal–ligand equilibria could not be studied using potentiometric methods. The iron binding constants of the deferoxamine analogs were determined by spectrophotometrically following the proton‐dependent exchange of iron with EDTA in the pH range of 4.0 to 6.5 and solving mass balance equations. The proton‐dissociation constants and the iron binding constants of the lipophilic deferoxamine analogs were comparable to those of deferoxamine. However, at physiological conditions, the iron‐binding complex of the most lipophilic butylamide derivative was slightly less stable and the succinamide derivative complex was slightly more stable. Like deferoxamine, the hydroxamate groups of the analogs were unhindered and free to form a 1:1 coordination complex with iron(III). Consequently, changes in aqueous solvation, conformation, and steric interference, imparted by the modifications at the terminal amino group of deferoxamine, may have affected the stabilities of the iron(III) complex and the efficiency of iron binding. © 2002 Wiley‐Liss Inc. and the American Pharmaceutical Association J Pharm Sci 91:1733–1741, 2002

A virtual node method for handling well bore boundary conditions in modeling multiphase flow in porous and fractured media

How to treat well boundary conditions is a difficult issue when formulating and coding a multiphase numerical reservoir simulator. The difficulty arises because the partial differential equation governing multiphase subsurface flow is of a mixed parabolic‐hyperbolic type. Maximum changes in primary variables and mass/heat fluxes occur at boundaries or well nodes. Consequently, these well nodes with small volume tend to be singular, leading to computational convergence problems. The conventional method of well treatment in geothermal or oil reservoir simulators is to use a sink/source term approach and to distribute flow rates by a potential or mobility allocation scheme for a multilayered well. However, this traditional method cannot handle a backflow problem, which may occur in a multilayered well in heterogeneous formations. This paper presents a “virtual node” method to handle a well bore either as a single node or several computational nodes screened and connected to many neighboring nodes for a multilayered well. The well bore can be vertical, inclined, or horizontal, and the well borehole node is treated in the same way as any other nonwell node for flow calculations. The solution at the well is then obtained by solving mass balance equations for the well node. It is shown that the new method provides a natural, physically consistent, and numerically efficient approach to handling well flow problems. Implementation of this new method for a three‐phase flow reservoir simulator is discussed, and three examples are provided.

Spatial variability in compartmental fate modelling

A new approach is presented which is designed to address the spatial heterogeneity of the environment in compartmental mass balance models of chemical fate in the environment. It rests on the assumption of chemical equilibration within one phase despite prevailing environmental heterogeneity. Composite D- and Z-values are derived from sub-unit specific environmental parameters and are used to solve mass balance equations which can be adopted essentially unchanged from existing compartmental fugacity models. With the resulting common fugacity value for each compartment, sub-unit specific concentrations and process rates can be calculated. The approach is illustrated using the QWASI lake model to calculate the fate of hexachlorobenzene in a hypothetical lake sub-divided in four distinct sub-units. The approach allows the subdivision of each compartment in a large number of sub-units with distinct environmental characteristics without substantially increasing model complexity. This is a necessary condition for linking fugacity models to geographical information systems.

Dynamic measurements of depth profiles of hydrogen implanted into graphite at elevated temperatures

Time variations of depth profiles of hydrogen retained in graphite during and after implantation of 3 keV H 2 + ions have been studied from dynamic measurements by means of elastic recoil detection at temperatures between 300 and 1000 K and at ion fluxes of 4 × 10 13, 2 × 10 14 and 4 × 10 14/ cm 2 s. It is shown that the maximum steady state concentration of hydrogens is about 6 × 10 22/ cm 3 at room temperature and is by a factor of 1.7 higher than the maximum static concentration after implantation. It is also shown that the depth profiles have the maximum at the projected range of implanted ions, the width of the profile broadens gradually inwards as the fluence and the temperature increase and the shapes of steady state profiles are almost the same as those of static profiles. By analytically solving mass balance equations for hydrogens, in which diffusion, trapping, ion-induced detrapping and recession of the surface are taken into account, the experimental steady state depth profiles are demonstrated to be well reproduced. The diffusion constants of hydrogen are determined as a function of temperature by fitting the calculated to the experimental depth profiles.

On the predictability of the 137Cs distribution in the Severn Estuary

The observed distribution of 137Cs resulting from known sources in the Severn Estuary permits the testing of simple predictive models for the one-dimensional distribution of non-conservative substances in an estuary. These models either use a known distribution of salinity to infer corresponding ones for other substances, or else directly solve mass balance equations utilizing previously determined dispersion coefficients. Both methods are shown to previously determined dispersion coefficients. Both methods are shown to provide results comparing favorably with observations. Stokes drift plays an important role in the circulation of the Severn. A modified densimetric Froude number, including this drift, is therefore used to estimate the circulation and stratification of the estuary. A new equation is used for calculating the 137Cs distribution from the observed salinity distribution. It generalizes previous formulations to permit variation of the net runoff with position along the estuary.