New Unstructured Mesh Water Quality Model for CoolingWater Biocide Discharges

Johannes Lawen & Huaming Yu & Georg Fieg &

Ahmed Abdel-Wahab & Tejas Bhatelia

Received: 12 January 2012 /Accepted: 24 April 2013 /Published online: 23 October 2013# Springer Science+Business Media Dordrecht 2013

Abstract A new unstructured mesh coastal water andair quality model has been developed that includesspecies transport, nonlinear decay, by-product formation,and mass-exchange between sea and atmosphere. Themodel has been programmed with a graphical user in-terface and is applicable to coastal seawater, lakes, andrivers. Focused on species conversion and interactionwith the atmosphere, the water and air quality modelfollows a modular approach. It is a compatible modulewhich simulates distributions based on fluid dynamicfield data of underlying existing hydrodynamic andatmospheric simulations. Nonlinear and spline approxi-mations of decay and growth kinetics, by-product for-mation, and joint sea–atmosphere simulation have beenembedded. The Windows application software includesfunctions allowing error analysis concerning mesh andfinite volume approximation. In this work, a submergedresidual chlorine cooling water discharge and halogenat-ed matter by-product formation has been simulated. Anerror analysis has been carried out by varying verticalmeshing, time-steps and comparing results based onexplicit and implicit finite volume approximation. Thenew model has been named 3D Simulation for Marineand Atmospheric Reactive Transport, in short 3DSMART.

Keywords Water quality model . Finite volumemethod .

Matrix reordering . Biocide . Chlorination of cooling water .

Halogenated organic compounds

1 Introduction

Discharges of cooling water into receiving water bodies arean environmental concern [1] since discharged biocides,such as residual chlorine, are often harmful to aquatic life[2]. Therefore, many cooling water discharge constituentsare regulated [3]. In order to assess their environmentalimpact, cooling water plumes can be simulated. Coastaltransport simulations and plume prediction models are oftendiscussed in scientific articles [4, 5] and constitute a signif-icant consultancy sector [6] in environmental impact assess-ment. Besides advective transport and dispersion, thedevelopment of hypochlorous, hypobromous acid, and ha-logenated matter plumes depends on reactive consumptionor accumulation and oxidants mass-exchange with the at-mosphere. The accurate approximation of the coastal waterbody, the incorporation of shoreline geometry, is importantfor a representative depiction of the simulated domain.Unstructured meshes are very suitable for complex domainshape approximation. The unstructured mesh allows an in-creased resolution at areas of importance, such as shorelinesand surrounding discharge outfalls, while the cell size canbe inflated for bulky areas of less importance, thus, savingcomputational time.

Several solely structured mesh and even closed formmodels, such as GEMSS [7] and Cormix [8], have beenused in environmental impact assessment. Heretofore, thesemodels neither allowed incorporating generalized kineticmodels beyond first-order decay nor generalized modelingof plume constituent mass-exchange with the atmosphere.

J. Lawen (*) :H. Yu :A. Abdel-Wahab : T. BhateliaTexas A&M University at Qatar, Texas A&M EngineeringBuilding, Education City,23874 Doha, Qatare-mail: johannes.lawen@qatar.tamu.edu

J. Lawen :G. FiegHamburg University of Technology, Schwarzenbergstr. 95,21073 Hamburg, Germany

Environ Model Assess (2014) 19:1–17DOI 10.1007/s10666-013-9370-6

1.1 Overview and Focus of the New Water and Air QualityModel

The water and air quality model introduced in this publicationcontains the following features:

1. The model utilizes unstructured meshes which allow asmooth increase of the cell density (the resolution) inareas of importance, thus achieving a better approxima-tion of shorelines, channels, and industrial outfalls.

2. Nonlinear conversion models or spline approximationsof laboratory measurement time series can be employed.Therefore, nonlinear by-product formation and fate canbe simulated.

3. Mass-exchange with the atmosphere has been embed-ded via Henry equilibrium coefficients. If the simulatedspecies occur naturally in the atmosphere (e.g., oxygen orcarbon dioxide), then a corresponding equilibrium con-centration can be set in the top layer of the water domain.If the simulated species does not constitute an air fraction(e.g., biocides such as chlorine), then calculating species-dependent equilibrium based mass-exchange at the sea–atmosphere interface requires simulating the species dis-tribution in the surface-near atmosphere. Otherwise,neglecting minor species accumulation (due to volatiliza-tion) in the surface-near atmosphere and assuming anatmospheric species concentration of zero (for e.g., chlo-rine) would correspond to a sea top-layer equilibriumconcentration of zero. This would result in solelydispersion-dependent volatilization regardless ofcomponent-specific volatilization behavior. Species vola-tilization requires, therefore, the simulation of its distribu-tion throughout the near-sea surface atmosphere. The 3DSimulation for Marine and Atmospheric Reactive Trans-port (3D SMART) can simulated the coastal water body’sresidual chlorine concentration coupled with theatmosphere.

4. Options integrated for error analysis concerning themesh dependency of results and allow a comparisonbetween explicit and implicit solution approaches.

5. Algorithms have been developed [9] which are opti-mized for flow field superimposed water and air qualitysimulations. With the available flow field information,the transport simulation’s solver uses the phenomeno-logical approach of fluid flow-guided matrix reordering,allowing the explicit calculation of stable implicitapproximations.

The model design approach is focused on water and airquality simulation including contaminant/species-specificdriving forces such as conversion and mass-exchange withthe atmosphere within a modular structure, with a separatehydrodynamic simulation. The 3D SMART has beenprogrammed as a Windows application in FORTRAN 90

with graphical user interface and result plotting functionsincluding interpolation. The developed tool furthers theestablishment of unstructured meshes in water and air qual-ity simulations. Complex kinetics beyond first-order decayterms can be depicted based on an automatic spline approx-imation of laboratory measured oxidants decay and haloge-nated compound formation. If measurement distributionsinclude all dependencies, all relevant parameters (e.g., tem-perature or salinity), then utilizing spline approximationsmight relieve the engineer or consultant from developing akinetic model. The simulation of excess oxidants mainlyconsiders the fate of hypobromous acid, since under seawa-ter conditions residual chlorine quickly converts to chloridewhile producing hypobromous acid [10]. The simulation ofthe species distribution is split into a hydrodynamic and asuperimposed reaction and flow field-driven advectionsimulation—a strategy relying on the prerequisite thatchanges of hydrodynamic properties since reactions arenegligible [11]. This prerequisite is certainly fulfilled forthe biocide concentration ranges operated at cooling cir-cuits. The coastal flow field computation in this work hasbeen carried out with the unstructured grid Finite VolumeCoastal Ocean Model [12–14].

1.2 Error Analysis—Horizontal and Vertical Meshingand Explicit Versus Implicit Approximation

In this work, an error analysis has been carried out, de-scribed in detail in Section 4, to identify error contributionestimates for the model parameter: (1) vertical layer distri-bution, (2) time-step, and (3) explicit versus implicit approx-imation. A submerged anoxic discharge has been simulatedwith 16 different mesh, time-step, and algorithm settings.The developed tool allows systematic analyzing of errorsbased on both horizontal unstructured mesh and verticallayer-grid resolution. Horizontal as well as vertical meshingcan be split into triangle sub-meshes and sub-layers, respec-tively, in order to obtain a comparable accurate referencesolution to quantify errors based on the finite volume do-main approximation. Also, the developed tool’s implicitmatrix reordering-based solver can produce reference solu-tions to allow estimating error contribution of usual explicit(time-wise) approximations.

1.3 Vertical Sub-Meshing

Furthermore, the study illustrates the possibility of increas-ing the vertical resolution of a water quality simulationcompared to the underlying hydrodynamic simulation bymeans of vertical sub-layering. This is relevant for constit-uents that exhibit significant vertical gradients due to vola-tilization. Other coastal ocean studies concerning volatilespecies, such as free residual chlorine, did not utilize small

2 J. Lawen et al.

boundary layers in the surface region [7]. The 3D SMARTcanadd several sub-layers in the surface region in order to accu-rately approximate the near-surface concentration profile.

2 Methodology

The structure of the modules is depicted in Fig. 1. The devel-oped tool (module G and C in Fig. 1) is suggested to beintertwined in a structure which contains a water and airquality model (G), atmospheric flow and dispersion (H), aswell as a hydrodynamic model (D). All share the same un-structured horizontal mesh, as shown in Fig. 1, from ameshing tool (E) like the Surface Modeling System fromAquaveo. Based on this mesh, a hydrodynamic tool (D) suchas the Finite Volume Coastal Ocean Model (FVCOM) is usedto compute for each time-step flow, dispersion, salinity, tem-perature, and surface elevation fields. Likewise, the atmo-spheric model (H), such as a k-ε model [15] in FLUENT,simulates velocity and dispersion fields for the atmosphere.Flow, dispersion, and surface elevation fields are fed into the3D SMART (G). If necessary, for kinetic (F) or marine lifemodels (I), temperature and salinity fields are transferred tothe 3D SMART (G) as well, in order to set individual reactionrates in each finite volume cell. The coastal hydrodynamic andwater quality simulation receive their source terms from aplant cooling circuit model (C). A cooling circuit mesh isgenerated with FLUENT-related meshing modules (A); theflow field inside the circuit is simulated with FLUENT (B)

and then fed into the cooling circuit biocide model (C). Thisprocedure has been examined by exporting the flow field fromFLUENT (B) to the cooling circuit biocide model (C) withTechplot. The advantage of this strategy is that all modelswhich consider biocide and by-product fate are integrated inone water quality model (G and C) which receives flow fieldsfrom three different fluid dynamic simulations: cooling circuit(C), coastal water body (G), and near-surface atmosphere (G).The two gray-shaded modules (C and G), cooling biocidemodel and water and air quality model (3D SMART), consti-tute the developed model which is presented in this work.

The 3D SMART’s transport model is a set of finite volumeequations (FVE). The FVE can either be derived from anadvection, dispersion, and reaction partial differential equation(PDE) or by directly balancing mass flow through the finitevolumes. The PDE is a balance of molar in and outflow of aspecies, j, along an infinitely small control volume. Terms fordispersive mass flow, conversion and convection are taken intoaccount. In and outflows at ri+dri are written at ri, the cell

center, by substituting n�ri þ drið Þwithn� rið Þ þ @ n

�rið Þ @ridri= .

Linearized reactions l for each species j are notated with stoi-chiometric, σjl, and kinetic matrix, kjl.

@cj r; tð Þ@t

¼Xi

@

@riD r; tð Þ @cj r; tð Þ

@ri� ui r; tð Þcj r; tð Þ

� �

þXl

σjlkjl cj r; tð Þ� �cj r; tð Þ

ð1Þ

or in finite volume method suitable notation

cj r; tð Þ ¼Z

1

dV

Xi

D r; tð Þ @cj r; tð Þ@ri

� ui r; tð Þcj r; tð Þ� �

dAi þXi

σjlkjl cj r; tð Þ� �cj r; tð Þ

!dt ð2Þ

(cj(s)) is a kinetic rate dependent on the componentconcentration. If the dependency of the rate on the concen-tration is given in the form of discretized data, then theconversion can be depicted with linear splines. The rate isupdated dependent on the time and space-wise localconcentration.

The FVCOM hydrodynamic model uses the shallowwater assumption-based primitive equation approximationsin the set of momentum, mass, and continuity balance.These equations may be denoted as it is common in thecorresponding hydrodynamic models [16, 17] with the threevelocity components u, v, and w [18]:

@u@t þ u @u

@x þ v @u@y þ w @u

@z � fv ¼ � 1ρ0

@P@x þ @

@z Km@u@z

� �þ Fu@v@t þ u @v

@x þ v @v@y þ w @v

@z � fu ¼ � 1ρ0

@P@y þ @

@z Km@v@z

� �þ Fv@P@z ¼ �ρg and @u

@x þ @v@y þ @w

@z ¼ 0:

ð3Þ

where x, y, and z are the east, north, and vertical axes. TheFVCOM furthermore uses a heat and salinity balance andcomputes the density ρ as a function of temperature andsalinity. P is the pressure, f the Coriolis parameter, g is thegravitational acceleration, Km is the vertical eddy viscosity,and Fu and Fv represent the horizontal momentum diffusionterms [18].

3 The 3D Simulation for Marine and Reactive Transport

The developed water and air model solves the transport PDEfor the water and air phase including exchange at the sea–atmosphere interface, hypobromous acid decay, and haloge-nated compound formation. New functions supporting erroranalysis are described in Section 3.5 and an implicit solveroptimized for flow field superimposed water quality com-putations is explained in Section 3.2. The explicit and

New Unstructured Mesh Water Quality Model 3

implicit form of the adaptive upwind finite volume equation,for all ctih 2 cth cells iy ∊ I in a layer h, can both be expressedwith:

ctþΔh �cthΔt ¼ P5

s¼1gths ctnl I ;sð Þh � cth

� �

�P3

s¼1

�v thsvth

eths 1� pths� �

cth þ pthsctnl I ;sð Þh

� ��

þ 1� pths� �

ctþΔh þ pthsc

tþΔnl I;sð Þh

� �1� eths� ��þ kth

2 cth þ ctþΔh

� �

ð4ÞThe logical control bit etihs 2 eths regulates whether advec-

tive flows through side s of triangle ih is approximated explicitor implicit. Dispersion through the five surfaces of one of theunstructured mesh’s triangle prism is quantified by gths ¼Δathsδ

ths Δrsvth� ��

where dtihs 2 δths is the dispersion coefficientvector of cell ih at surface s; Δatihs 2 Δaths is the finite volumecell surface vector; Δrtihs 2 Δrths is the inter-centroid distancevector and vtih 2 vth contains all cell volumes for a layer h.�vtihs 2 �vths is the volume flow vector for cell ih with respect toside s. The closed form kinetic model-based constant, or splineconstant,ktih 2 kth corresponds to a certain concentration leveland, as the case may be, to further parameters, such as temper-ature, which are assigned to cell i and time level t. pihs 2 πhs

regulates what percentage a concentration value of a certaincentroid (the concentration of cell i itself or of its neighbor)factors into the flow approximation. πihs is the adaptive upwindcontrol bit which functions to approximate flow calculation to0 % based on the mass-receiving centroid and to 100 % basedon the donating centroid (except at open boundaries). The firstrow in Eq. 4 represents dispersive exchange. The remainder ofthe equation contains the approximation of horizontal advec-tion and conversion. Flow field, dispersion, surface elevation,

salinity, and temperature arrays can be fed to the program byany hydrodynamic simulation which stores data either on thetriangle centroids, nods, or centers of the finite volume sur-faces. For the error analysis in Section 4, data from theFVCOM have been fed into the 3D SMART.

3.1 Boundary Conditions

Ten boundary conditions have been embedded into the 3DSMART. Table 1 lists the differential relations for the bound-ary conditions and the approximations considering the finitevolume equation system. Boundary condition approximationsnotated at ihs hold for one cell ihs. Vector type notatedapproximations (such as cth and gths) hold for all active cellsin layer h. The horizontal open boundary condition, presentedin Table 1, defines the gradient at the boundary to be zero forboth water and air. This suppresses dispersion but permitsadvection through the open boundary.

The no-flux boundary condition assures that neither dis-persive nor advective flux passes through a solid boundary.At the sea and air boundary layer, dispersion is replacedwith a Henry coefficient equilibrium assumption. This re-quires that hypobromous acid is simulated throughout thenear-surface atmosphere. If the species occurs in the atmo-sphere (e.g., O2 or CO2), then the corresponding equilibriumconcentration (depending on salinity and temperature) is setin the entire top layer of the water body and the simulationof the near-surface atmosphere can be omitted. Both of theseequilibrium boundary conditions apply for the entire bound-

ary layer and are hence indicated in vector notation, ctþΔh , for

all cells in the layer. The vertical open boundary conditionfor the atmosphere extrapolates the vertical gradient to theother side of the boundary which permits dispersion and has

Hydrodynamic Coastal Model(E.g. FVCOM)

Water & Air Quality Model1. Preprocesssub-mesh & water quality boundary conditions

2. Species fate simulation3. Visualization

Coastal Unstructured Meshing(E.g. SMS)

Atmospheric Model(E.g. FLUENT)

Marine Life Model(Oxidants and halogenated organic matter toxicity & species decay)

Seawater Biocide Model(Providing kinetic rates fordecay & halogenated byproduct formation;)

Cooling Circuit Biocide Model(Decay & halogenated byproduct formation)

Hydrodynamic Cooling Circuit Model(E.g. FLUENT)

Cooling Circuit Meshing(E.g. FLUENT)

Flow, disp. & temp. fields

Hydrodynamic fieldsContaminant source term

Hydrodynamic source term

(a) (b)

(c)

(g)

(e)

(f)

(d)

(h)

(i)

Fig. 1 Modular modelstructure of meshing,computing fluid dynamics, andair and water quality. Thehydrodynamic and theatmospheric models (D and H)compute flow and temperaturefields for the water and airquality model (G). The coolingcircuit biocide model (C)receives flow fields fromFLUENT (B). All threesimulations (D, H, and G)utilize the same horizontaltriangle mesh generated by theunstructured meshing tool

4 J. Lawen et al.

been chosen to avoid an insulating functioning of the con-dition since dispersion is the dominating transport mecha-nism along the vertical extension. Intakes can beapproximated by two different boundary conditions: (1)mass conservation between intake inflow and the flowadvected through the surfaces of the cell is assumed.Approximated for finite volumes, this is equivalent tosetting all volume flows through the cell surfaces to zero.This has the effect that the concentration of the cell willbe evened by dispersion to the levels of the surroundingcells. (2) Alternatively, the intake cell can be set to aconstant value which has been measured. This boundarycondition can also be utilized when measurements havebeen taken at some locations and the ensuing plume is ofinterest. Therefore, this boundary condition can be used fordischarges, intakes, and calibrations with measurementvalues. Table 1 also lists a source term boundary conditionwhich can be used for discharges as an alternative to setting aconstant value. Contrary to the adaptive upwind principle,advection through the open boundaries is always calculatedbased on the cells which are located at the boundary. This isembedded by including the boundary condition information inthe adaptive upwind control bit πihs.

3.2 The Semi-implicit Fluid Flow Adaptive Solver

The developed software can simulate species distributionsbased on time-wise explicit or implicit finite volume equations.The fluid flow adaptive solver allows solving a semi-implicit

adaptive upwind system solely by means of matrix reordering.εihs is the logical bit which sets the implicit partition of time-wise species concentration approximation and is usually at 0.5to assure an even averaging between explicit and implicitapproximation. Assuming implicit or partially implicit approx-imation, the finite volume equation system (FVES) corre-sponds to a matrix form MctþΔ ¼ b with M containing the

coefficients of concentrations at subsequent time level ctþΔih .

This system could be solved entirely by matrix reorderingunder the condition that no circulations are given in the flowfield. Unfortunately, coastal simulations contain circulations.Therefore, approximation and solver have been integrated, aspublished in a separate work [9], to generate a partially explicitand partially implicit approximated FVES. This is solvable bymatrix reordering but keeps the most volume flows (and rela-tively stability affecting flows) in implicit approximation. Thematrix reordering is carried out by calculating cells sequential-ly along the direction of the fluid flow. Due to the adaptiveupwind principle, implicit approximated concentration values,

ctþΔih , can be isolated and calculated during fluid flow directionadaptive ordered computation. In case of circulations, relative-ly slow flows between relatively large cells are approximatedexplicit and εihs is equal to 0. For the explicit solver, εihs isalways equal to 1.

3.3 Hydrodynamic Field Interpolation

Hydrodynamic field data, velocity components, and, if nec-essary for the calculation of Henry equilibrium coefficients

Table 1 Ten boundary conditions have been embedded into the de-veloped model. The unit surface normal n(r) stands perpendicular onthe boundary of the domain ∂Ω(r) with the gradient ∇nc(r) beingnotated with ∂c/∂n. z is the vertical extension and indicated below in

boundary conditions at the air–water interface intf or at the verticalupper boundary of the atmosphere. The outer side of the vertical upperboundary is being referred to with 0−

Boundary condition with n?@Ω FVE approximation 8ihk corresponding to the BC

Horizontal open 0 ¼ @c@n n¼0j pihs ¼ 0 ^ gtihs ¼ 0

No flux 0 ¼ uc� D @c@n n¼0j v

� tihs ¼ 0 ^ gtihs ¼ 0

Sea boundary layer cw ¼ catkeq

^ @c@z

atintf

¼ 0 ctþΔ

hsea¼ ctþΔ

hairvthair

þctþΔhsea

vthseakeqvthair

þvthsea^ gthseas ¼ 0

Air boundary layer cw ¼ catkeq

^ @c@z

atintf

¼ 0 ctþΔhair

¼ ctþΔhair

vthairþctþΔ

hseavthsea

vthairþ

vthseakeq

^ gthairs ¼ 0

Sea boundary layer with constituent in atmosphere cw ¼ catkeq

^ @c@z

atintf

¼ 0 ctþΔ

hsea¼ cat

keq^ gthseas ¼ 0

Vertical air BC at atmospheric top layer D @c@z z¼0� ¼j D @c

@z z¼0j ctqþ1 ¼ gtqs ctqs � ctq�1

� �þ ctq

Conservative intake (intake inflow�V int = advected inflow)

V�int

¼ ∯T

uðrÞ � nðrÞð ÞdTðrÞP3s¼1

�v ihs ¼ 0

Measured value m(r,t) c=m(r, t) ctþΔih ¼ mt

ih

Discharge source term s(r,t) c=FVE+s(r, t) ctþΔih ¼ ctþΔ

ih þ stihΔvtih

hsea top-layer water, hair ground-layer atmosphere, n(r) unit surface normal, T(r) cell interfacial surface with neighboring cells, FVE finite volumeequation, cw concentration in water, cat concentration in air (O2 or CO2), keq equilibrium coefficient, q top layer in atmosphere

New Unstructured Mesh Water Quality Model 5

and kinetic models, then also temperature and salinity fieldsare transmitted from the hydrodynamic simulation to the 3DSMART. Ideally, velocity component field data are stored atthe center of triangle prism surfaces, since the simulationrelies on the balance of mass flow through the volumesurfaces. Preferable velocity components at the surfacesare volume and mass flow conserving for each cell andbetween neighboring cells. However, velocity componentfield data from the FVCOM model are generated at thetriangle centroids (surface elevations are stored at trian-gle nods). Therefore, the velocity components at the cellsurfaces must be interpolated. The FVCOM flow field isfurthermore not volume flow and mass flow conservativefor a single cell. Three interpolation procedures havebeen embedded:

1. Taking the velocity components of a triangle centroidfor each surface of the cell.

2. Linear-weighted interpolation between two neighboringcell centroids.

3. Averaging the velocity values corresponding to the twotriangle nods which form a triangle side; velocity valuesat triangle nods are calculated by averaging the valuesof all triangles which share the considered nod.

The first procedure is the most primitive; it assures fullmass flow conservation for a certain cell (except for cells ata solid boundary), inflow equals outflow, while mass flowconservation is not achieved considering inter-cell mass-exchange (flow leaving a donating does not equal the inflowinto the neighboring receiving cell). The second interpola-tion procedure assures mass conservation between neigh-boring cells but does not achieve mass conservationconsidering the in- and outflow balance of a cell.

Again, preferably fully mass flow conserving hydrody-namic field data corresponding to the center of cell surfacesare utilized. But this is a challenging prerequisite consider-ing available 3D unstructured mesh dynamic coastal oceanmodels.

3.4 Atmospheric Flow Field Model

The atmospheric or planetary boundary layer (PBL) is de-fined as the layer in which the Earth’s surface affects theatmosphere through momentum, heat, and moisture ex-change occurring over time scales of a few hours to less thana day [19]. In general, in the top of the PBL, turbulent motionis not significant compared to the lowest part, roughly 10 %,of the PBL. Momentum, heat, and moisture vertical fluxes donot vary by more than 10 % above the surface layer [20].Similarity theories such as the Monin–Obukhov scaling the-ory provide the flux relations for velocity, potential temper-ature, and specific humidity (for oceanic environmentswhere it is important to account for transfer of water vapor).

These flux relations generate velocity, temperature, and hu-midity profiles that provide boundary and initial conditionsfor the atmospheric boundary layer (CFD) model. The k-εturbulence model (or its variations, e.g., RNG k-ε), based onthe Reynolds Averaged Navier–Stokes equations, can re-solve the atmospheric boundary layer. Density variationsare introduced into the momentum equation using theBoussinesq approximation and appropriate buoyancy termsincluded in the k and ε equations. Such turbulence modelshave been successfully used for atmospheric flows but, asmentioned above, they require modification of the form orvalues of the coefficients [21–23]. Further, one should alsoinclude modified momentum equations by the addition of theCoriolis and centrifugal forces. Modified k and ε equationbased models simulate velocity and dispersion fields for theatmospheric boundary layer under stable, neutral, or unstablethermal stratifications.

3.5 Investigating Result Dependency on Horizontal,Time-Wise, and Vertical Meshing

An error analysis can be carried out by comparing moreaccurate system approximations with less accurate refer-ence approximations. However, it should be differentiat-ed between error analysis, sensitivity analysis (comparingdisturbance of input and output), and uncertainty analysis(when different system properties appear to not be pref-erable among each other). Figure 2 has been split into 16sub-triangles. A triangle can be split into 4n sub-triangleswith the computational load increasing analogously withthe factor 4n. To achieve an error estimate for solution Yi,it can be compared with a solution of a higher order ofaccuracy with Errori � Yaccurate � Yið Þ Yaccurate=j j [24] ifYaccurate � Ytruej j � Yi � Ytruej j . Simulation parameterssuch as the horizontal mesh resolution, number of verti-cal layers, time-step, and differential equation approxi-mation are considered to fall under the category of erroranalysis in this work. The accuracy depending on thehorizontal resolution can be evaluated by splitting tri-angles of a mesh into sub-triangles as shown in Fig. 2.Also, vertical layer thickness and number can be variedby splitting a layer into sub-layers, as in Figs. 3 and 4, toinvestigate its influence on vertical dispersion.

The 3D SMART allows effectively generating referencesolutions for the following model parameters:

1. Horizontal meshing—the triangles of the horizontal tri-angle mesh can be split into sub-triangles for compara-tive simulations;

2. Vertical meshing—vertical layer can be split into sub-layer for comparative simulations or to increase theresolution;

3. Time-step can be varied.

6 J. Lawen et al.

An error analysis has been carried out and is de-scribed in detail in Section 4. In Fig. 4, investigatingsolution accuracy depending on the number of layershas been illustrated for dissolved oxygen. Consideringonly vertical dispersion along z, biological sink, source(or reactions due to other reactants), and hypotheticalquasi-steady state, the simplified differential relation

Ddispd2co2ðzÞ dz2 ¼ �kbioco2ðzÞ�

has been solved numeri-cally for various layering configurations. The number oflayers might also affect the accuracy considering theatmospheric part. The interaction of atmospheric andsea plume are compared in Fig. 5, based on a numericalexperiment which shall illustrate the strong effect of theatmosphere and sea boundary layer thickness.

Fig. 2 The 3DSMARTsplits each triangle of themeshA (pink) into 16 sub-triangles (green) to evaluatemesh dependency as demonstrated at DohaBayincluding the artificial islands Lusail (north) and The Pearl Qatar (center)

5.597 5.598 5.599 5.6 5.601 5.602 5.603 5.6042.7608

2.7609

2.761

2.7611

2.7612

2.7613

2.7614

-10

-5

0

5

Easting/[105 m]

Dep

th/[m

]N

orth

ing/

[106

m]

Fig. 3 Black triangle prismsrepresent water in a channel.The cyan triangle mesh layeremphasizes the splitting of alayer into two sub-layers. Themagenta triangle mesh layer isthe surface between sea and airwhich adapts to surfaceelevation due to tidalmovements. This has beengraphically emphasized bysuperimposing a sinusoidalsurface elevation onto the redsurface mesh. The yellow layersrepresent the atmosphere

New Unstructured Mesh Water Quality Model 7

3.6 Integrated Cooling Circuit Simulation

Biocide conversion and by-product formation occursalready in the cooling circuit. The practicability of anintegrated plume fate and cooling circuit simulation hasbeen examined by utilizing the water quality model ona heat exchanger-type geometry. Both unstructured and

structured type meshes are possible. Whereas unstruc-tured meshing has been utilized for the water qualitymodel, a structured mesh has been utilized for the heatexchange simulation. The model structure has been suc-cessfully validated by exporting the FLUENT hydrody-namic field data to the water quality model viaTechplot. The data transmission is less complicated for

0 1 2 3 4 5 6 7 8-10

-8

-6

-4

-2

0

2

4

DO/[mg/L]

vert

ical

ext

ensi

on, d

epth

/[m]

Vertical Mesh Illustration and Result Dependency on the Vertical Resolution

at vertices elevated surface approximation

sub-layer

10 cells with LU decomposition20 cells with LU decomposition10 cells, as parabolic PDE, explicit20 cells, as parabolic PDE, explicitanalytical solution

Fig. 4 A 2D view showing thescheme of examining theinfluence of the verticalmeshing onto a vertical resultdistribution. Green plus signsand circles show results for avertical resolution of 0.5 m,whereas red plus signs andcircles correspond to 1 m. Thegreen layers emphasize sub-layers in the model which allowexamining the resultdependency on the verticalresolution

atmospheric transport

sea transport

Fig. 5 Numerical demonstration of coupled sea–atmosphere simula-tion of a tracer plume in lower atmosphere layers. The volatile tracer istransported via Henry coefficient type coupling from the eastwardheading sea plume into the atmosphere. Inside the atmosphere, the

biocide is transported with southward wind. The Henry coefficient typecoupling of air and sea requires a relative small size of the near-surfacelayers. Fifty percentage of the maximum in the distribution is themaximum of a color scale with violet indicating the absence of tracer

8 J. Lawen et al.

the cooling circuit than for the coastal water body dueto two aspects: (1) due to the simpler physics, FLUENTcan produces mass conserving flow fields compared tothe mass conservation violating flow fields of theFVCOM; (2) simple steady-state hydrodynamic simula-tions can be meaningful for the cooling circuit whereascoastal simulations must be transient due to tidal dy-namics. In Fig. 6, flow field superimposed speciestransport is simulated within heat exchanger-type geom-etries. Dynamic biocide conversion, in case of unsteadybiocide dosing, can be simulated superimposed on asteady-state flow field.

3.7 Kinetics and Mass-Exchange with the Atmosphere

The nonlinear hypobromous acid reaction with organicmatter (to products such as TBM) can be simulated byclosed form nonlinear models or a spline approximationwhich fits itself to laboratory measurements.

An individual first-order decay constant correspondingto a certain concentration interval is selected for each cellfor each time-step. If no suitable closed form conversionmodel is available, a spline approximation which fits itselfto laboratory measurements can be employed. This maybe presented for the example of TBM in Table 2. Thespline fitting (Fig. 7) can be made dependent on a varietyof factors including temperature—otherwise it is run withdata which have been taken at a temperature range whichcorresponds to the simulated case. Interaction with theatmosphere is taken into account based on Henry coeffi-cients. If the constituent is present in the atmosphere, e.g.,oxygen or carbon dioxide (and sink or source function ofthe sea is negligible for the atmosphere), then a tempera-ture and salinity field-dependent equilibrium concentrationdistribution is assumed in the water boundary layer at thewater–air interface. If the constituent is not present in theatmosphere, e.g., free residual chlorine in case of biocidedischarge, then simulating the surface-near atmosphere isnecessary. In Section 4 an error analysis is carried out forhypobromous acid transport including a Henry equilibriumboundary condition at the sea–atmosphere interface. Atthe bromide-rich seawater conditions in the Arabian Gulf,free residual chlorine will form hypochlorous acid whichwill react further to a chloride and hypobromous acid[10]. The simulation will therefore consider excess oxi-dants (hypobromous acid). Hypobromous acid hasaccording to Sander [25], who cites Blatchley et al. [26],a Henry coefficient of kpxH;inv ¼ 0:0295� 0:0051atm which

is equivalent to kpxH;inv ¼ pg xa ¼ ρH2O

�MH2OkHð Þ= which

yields at T=298.15 K kH=1,932±334 M/atm. Sincehypobromous acid volatilization reduces the plume size,it is safer to overestimate the solubility. Hence, kH=1,932+334=2,266 M/atm shall be assumed which is inthe dimensionless ratio of air and gas concentrations keq=

Fig. 6 Biocide transport superimposed on flow field inside a heatexchanger simulated with adaptive upwind and adaptive solver;0.5 m/s inlet velocity into 1.5×2 m-sized heat exchanger approximatedwith a cell size of 1 cm

Table 2 Example forlaboratory measure-ments for kinetic data towhich the simulationcreates a splineapproximation

Timemin½ �

Residual chlorineg m3=½ �

TBMmol m3=½ �

0 3.345 6.750E−06

0.5 2.352 5.395E−04

1 1.53 6.780E−04

6 1.13 7.317E−04

18 0.868 7.396E−04

24 0.753 7.714E−04

48 0.474 7.921E−04

72 0.211 8.009E−04

96 0.103 7.999E−04

120 0 8.013E−04

168 0 8.019E−04

Fig. 7 Decay of residual chlorine according to laboratory measure-ments (blue points) and simulation (red line)

New Unstructured Mesh Water Quality Model 9

cw/cat=kHRT=55,403 where R is the ideal gas constant.Temperature and salinity dependencies of the equilibriumcoefficient are taken into account with

keq ¼ k*eqeΔsolvH

R

1

T� 1

T*

� �ð5Þ

[25] where ΔsolvH is the enthalpy of solution. Dewulf etal. [27] identified that, in general, volatility increases withan increase in salinity:

ln cat cw=ð Þ ¼ a

Tþ bcsalin þ c ð6Þ

Fig. 8 Water quality model interface showing a section of the mesh corresponding to the simulated area. The submerged discharge can be recognized inan area of high resolution

5.573 5.5735 5.574 5.5745 5.575 5.5755 5.576 5.57652.7559

2.7559

2.756

2.756

2.7561

2.7561

2.7562

2.7562

2.7563

2.7563

-10

0

10

Northing/[10

6 m]

Easting/[105 m]

Dep

th/[m

]

Discharges at left channel boundary.

Varying surface elevation.

Fig. 9 3D view of the triangle prism, stacked in layers, and horizon-tally assembled to the computational domain. The cyan layer shows theutilized sub-layer. High resolution decreases with increasing distancefrom discharges. Vertices of the magenta triangle layer are individually

elevated depending on tidal elevation. In this figure, a sinusoidalsurface elevation has been added to emphasize the elevation adaptionof the surface layer. The changing elevation is recognizable in theupper boundary

10 J. Lawen et al.

a is a negative constant whereas b and c are positiveconstants and csalin is the salt concentration in gram perliter.

Since, as stated above, mixing zone evaluation shall tendto underestimate volatilization, it is safer to use equilibriumdata originating from low salinity conditions [26].

3.8 Interface, Visualization, and Result Analysis

The software has been programmed in FORTRAN 90 in-cluding Windows application user interface and visualiza-tion. Distributions can be visualized for single layer oraverage concentrations of a number of layers. Distributions

Fig. 10 Plume in percent biocide after 2 h with 16 layers, a time-step of 0.3 s., using an explicit approximation

Fig. 11 Normalized absolute error estimate distribution, the difference between the 15- and the 16-layer distribution

New Unstructured Mesh Water Quality Model 11

can be shown in tripaving without interpolation, like inFigs. 10 and 11, or with interpolation like in Fig. 5. The typeof interpolation can be chosen among: linear triangle interpo-lation, where the triangle centers are Delaunay assembled to acomplementary triangle mesh, followed by linear interpola-tion inside the triangle, or different types of inverse distanceinterpolations [28, 29]. Geographical UTM coordinates areindicated as abscissa and ordinate axes. Governmentally reg-ulated mixing zones can be depicted as circles around thedischarge, as shown in Figs. 5, 10, and 11. The permittedradius to define the mixing zone depends on local environ-mental regulations, e.g., the ministry of environment in Qatarpermits 100 m for constituents such as residual chlorine. Theradius of a mixing zone can be set individually for eachdischarge. Zones where boundary values exceeded can bevisualized by an isoline plot to mark the border of a criticalzone. The user can therefore easily identify whether the zonewhere the boundary concentration exceeded lies only withinthe mixing zone or extends beyond the border of the mixingzone, violating regulations. The maximum concentration atthe boundary of the mixing zone is indicated in the left uppercorner as shown in Fig. 10.

4 Case and Error Analysis

An oxidant (hypobromous and hypochlorous acid) distributionemanating from several process water outfalls dischargingchlorinated cooling water has been simulated to carry out anerror analysis concerning vertical meshing, time-step, andexplicit approximation. A coastal section at the shores of theArabian Gulf, as shown in Fig. 8, has been approximated witha horizontally unstructured mesh with vertically structured

layers. The channel in Fig. 8 is depicted in 3D view inFig. 9. The mesh has 15 layers of 1 m (layers 1 to 8) and2 m thickness, dependent on the water depth. The layers varyin thickness to adapt to tidal movements. Horizontally, 12,784triangles assemble one layer. Cells in deeper regions are inac-tive in order to adapt to the bathymetry. For the cases depictedin Figs. 10 and 11, a discharge residual chlorine concentrationof 0.1mol/l has been set. Hypobromous acid volatilization intothe atmosphere has been taken into account by setting a Henryequilibrium-based concentration in the top layer. Several sim-ulations with different settings concerning vertical mesh, time-step, and approximation have been carried out for the erroranalysis. The following parameters have been varied:

1. Vertical resolution—the top layer has been split into twosub-layers.

2. Time-step—the time-steps 0.1, 0.3, 0.4, and 1 s havebeen employed.

3. The time-wise momentum of the finite volume equationhas been varied between explicit and implicit to evalu-ate the error due to explicit approximation.

The discharge temperature is 41 °C with an ambienttemperature of 34 °C. The 3D SMART and the FVCOMhydrodynamic model, which provided flow field and otherhydrodynamic field data, used the same mesh. The cell sizeis reduced in areas of importance to accurately depict coastaland outfall geometries. Close to the discharge, the horizontalcell size is only 1 m.

4.1 Test Results

Figure 10 shows the results for a domain approximationwith 16 layers. The distribution corresponds to the vertical

Table 3 Maximum concentration at the boundary of mixing zone in percentage

Time-step/[s] Explicit, 15 layer Explicit, 16 layer Semi-implicit, 15 layer Semi-implicit, 16 layer

0.1 27.72161 27.12886 27.71439 27.13478

0.3 27.71832 27.13227 27.71745 27.13224

0.4 27.71549 27.13290 27.71804 27.13099

1.0 27.71404 27.13290 27.72060 27.12890

Sur

face

ele

vatio

n/[m

]

Time/[days]

Fig. 12 Forty-eight-hour calibration of tidal simulation (black line) with tidal measurements (red crosses)

12 J. Lawen et al.

average along the water column. The results of the differentcases are compared in Table 3. Figure 11 shows the differ-ence between the average concentration of the simulationswith 15 and 16 layers.

The hydrodynamic simulation has been calibrated withtidal data obtained from the Environmental Study Center ofthe State of Qatar.

The simulated tidal elevations show good agreement(Fig. 12) with the measured data. Tidal dynamics have asignificant effect on the plume direction. Tidal effectsare taken into account in the simulation. However, thesmaller the scope of consideration, the smaller is theinfluence of tidal dynamics. The considered mixingzone of 200 m is therefore mainly governed by currentsinduced by the discharged flow while the tidal influenceis small at this scale. The vertical size of the domain isadapting itself to the tidal surface elevation. The layerthickness in the mesh is smoothly varying and incre-mentally changed.

Also, the horizontal velocity components (u and v)have been calibrated to measurement values from sta-tions located within the simulated domain and aredepicted in Figs. 13 and 14.

The order of magnitude of the measurement values inFigs. 13 and 14 and the simulation show good agreement.

The difference between a 15- and a 16-layer approxima-tion has been visualized in Fig. 11 which can function as aspatial and dynamic absolute error estimate distribution. Thetool is able to plot the difference between two simulateddistributions. Depending on the context, this can visualizethe error, the uncertainty, or the sensitivity. In Table 4, error

estimates are shown for the maximum concentration at theboundary of the mixing zone, shown in Table 3. The errorestimate, error A in Table 4, concerning the finite time-step,has been calculated based on Errori≈ |(Yaccurate−Yi)/Yaccurate|.The simulation with a time-step of 0.1 s is used in thisequation as the accurate solution EA whereas the less accu-rate solution is that with a time-step of 1.0 s. For both, the16- or the 15-layer case, it resulted in similar errors: 1.50×10−2 % for the 16 layers and 1.94×10−2 % for 15 layers.

The error estimate, error B in Table 4, concerning thebias of explicit approximation, has been calculated byusing, as a comparison, the semi-implicit approximation(Section 3.2) in which not less than 80 % of all inter-celladvective volume flows that have been evenly averagedbetween explicit and implicit approximation. Additionally,the remaining <20 % explicit part contains only veryslow and, hence, less system determining volume flows.The largest of all explicit approximated volume flows isbelow 25 % in size compared to the largest of the semi-implicit approximated volume flows. The semi-implicitapproximation has therefore been considered to be asufficient approximation of the ideal case in which eachinter-cell volume flow receives a 50 % implicit approx-imation. Considering the simulation with 15 layers, theerror estimate is 2.61×10−2 % for a time-step of 0.1 sand 2.37×10−2 % with a time-step size of 1.0 s.Employing a time-step size of 1.0 s on the explicitapproximation yielded stable results in this simulation,but showed emerging instability in other parts of thedomain during other simulations, indicating the commontime-step size limitations of explicit approximations.

Vel

ocity

/[cm

s-1]

Time/[days]

Fig. 13 Correlation of eastern u velocity component (red crosses) with the simulated velocity component (black line)

Vel

ocity

/[cm

s-1]

Time/[days]

Fig. 14 Correlation of northern v velocity component (red crosses) with the simulated velocity component (black line)

New Unstructured Mesh Water Quality Model 13

The error estimate, error C in Table 4, concerning thelayer resolution close to the surface has been calculated bycomparing the 15-layer simulation with the simulation in-cluding two sub-layers in the top layer. Table 4 showsselected examples for error C, for the explicit solver with atime-step of 0.1 and 0.4 s, and for the semi-implicit solverfor a time-step of 0.3 s.

5 Discussion of the Error Analysis

Table 4 shows estimates for error contribution due to verti-cal meshing, time-step, and implicit/explicit approximation.

5.1 Time-Wise Mesh Increment

Considering the explicit solver with increasing time-step, a slightly lower concentration at the boundaryhas been expected, considering that the explicit solvercalculates flows solely based on previous concentrationvalues corresponding to present time-steps. A biaswhich would grow with larger time-steps. While thistrend was confirmed in some simulations, presentedcases did not correlate with this assumption. We specu-late that loss of biocide into the atmosphere is equallyinhibited by larger time-steps and can offset the men-tioned bias. As expected, some explicit discharge simu-lations became instable with a time-step size of 1.0 s.

5.2 Sensitivity Considering Explicit or Semi-implicitApproximation in the 3D SMART

The errors of explicit and implicit solver do not showdifferent orders of magnitude. Comparing the explicit andsemi-implicit algorithms gives not necessarily an estimate ofthe uncertainty due to an explicit approximation. But theuncertainty due to approximating fluxes at t to calculate

future concentrations, as it is the case for explicit approxi-mations (with εths ¼ 1 in Eq. 4), can be assessed by means ofa semi-implicit approximation which averages approximat-ed flows between t and t+Δ.

5.3 Vertical Increment Along the Water Column

A trend can be observed by comparing the simulations with15 or 16 layers. Having a smaller increment at the surfaceyielded a slightly smaller biocide concentration. Furthercases would be necessary to calculate the error due to finitevertical layering.

Figure 11 illustrates a spatial and dynamic error distribu-tion. The distribution shows that only the error at the loca-tion of the highest constituent concentration at the boundaryof the mixing zone might be relevant when identifyingwhether mixing zone regulations are met. Taking this intoaccount might allow indicating lower error estimates sincethe absolute error at one location will most probably besubstantially lower than the highest error of the entiredistribution.

The above covers most of the error contribution dueto (time-wise) explicit or implicit approximation anddue to the space time-wise meshing of the water andair quality model excluding the horizontal mesh resolu-tion. Including horizontal meshing into the error analy-sis requires varying the horizontal resolution bysplitting the triangles in 4, 16, 64, or 4n sub-trianglesas depicted in Fig. 2. An error analysis for the entirewater and air quality model would require also exam-ining error contribution due to the spatial momentum ofthe algorithm and comparing algorithms of differentaccuracy.

Analogous to the modular approach of this water and airquality model, a disturbance array of the hydrodynamicsimulation can be used to simulate the resulting disturbancein the water quality distribution.

Table 4 Error estimates related to the boundary of the mixing zone

Model parameter Error estimate in %, 15 layer Error estimate in %, 16 layer Reference solution

Error A 2.73×10−2 1.50×10−2 Δ=1.0 compared with Δ=0.1 s, explicit

Error B 2.61×10−2 2.18×10−2 Δ=0.1 s, semi-implicit

Error B 3.14×10−3 1.11×10−4 Δ=0.3 s, semi-implicit

Error B 9.20×10−3 7.04×10−3 Δ=0.4 s, semi-implicit

Error B 2.37×10−2 1.47×10−2 Δ=1.0 s, semi-implicit

Error C 2.18 Δ=0.1 s, explicit, 16 layer

Error C 2.15 Δ=0.4 s, explicit, 16 layer

Error C 2.16 Δ=0.3 s, semi-implicit, 16 layers

Error A error estimate due to finite time-step size, error B error estimate due to the bias of the time-wise momentum for an explicit approximation,error C error estimate due to finite layer size in the approximation of near-surface waters, Δ time-step size

14 J. Lawen et al.

The errors are very small compared to the magnitude ofthe constituent over the extension of the mixing zone.Therefore, considering the two above-mentioned aspects ofspatial and time-wise meshing, the results are reliable. Thefluid flow adaptive solver allows assessing the bias of anexplicit approximation due to the bias of the time-wisemomentum of an explicit finite volume equation.

6 Conclusion

The presented simulations show that errors due to time-stepsize and explicit approximation are very small; as evidence,no error above 2.73×10−2 % has been found for these errorcategories. With 2.18 %, the most significant error magni-tude has been found to be connected to the near-surfacevertical meshing where the biocide exchange with the atmo-sphere takes place.

The vertical sub-layering and the semi-implicit ap-proximation are suitable to function as comparisons inan error analysis within the validation process. Validat-ing the sufficiency of a vertical layer resolution isespecially valuable considering the biocide volatiliza-

tion. The visualization allows several different interpo-lation techniques and color scales. Plots allowsuperimposing, subtracting, and visualizing the resultsof different scenarios to analyze errors, uncertainties,or sensitivity. The 3D SMART’s visualization is opti-mized for the analysis of water and air quality resultsconcerning their compliance with regulated boundaryvalues and mixing zone sizes. Also, it has been illus-trated that indicating the maximum of a dynamic abso-lute error estimate distribution (Fig. 11) might not benecessary; only the maximum error at the boundary ofthe mixing zone is relevant when considering compli-ance with mixing zone regulations.

The modular approach of separate hydrodynamicmodel and water and air quality model suggests evalu-ating uncertainties separately by feeding disturbed flowfields from the hydrodynamic simulation into the waterand air quality model to evaluate the impact on theplume shape.

Acknowledgments This publication was made possible by NPRPgrant # 29-6-7-39 from the Qatar National Research Fund (a memberof Qatar Foundation). The statements made herein are solely theresponsibility of the authors.

Appendix A

Table 5 Symbols for the partialdifferential equation and bound-ary conditions

Symbol Description Unit

n�

Molar mass flow mol/s

cj(r,t) Concentration of species j at r and τ mol/m3

j Species index –

ri ∊ r Spatial location vector m

τ Time s

D(r, τ) Dispersion coefficient m2/s

ui(r, τ) ∊ u(r, τ) Velocity component vector m/s

l Reaction index –

σjl ∊ Σ Stoichiometric coefficient matrix for species j and reaction l –

kjl(cj(r, τ)) ∊ K Kinetic coefficient matrix dependent on cj(r,τ) 1/s

nðrÞ?@ðrÞ Unit surface normal stands perpendicular on the boundary m/s

V�int Intake volume flow m3/s

cw Equilibrium concentration in water mol/m3

cat Equilibrium concentration in air mol/m3

keq Henry coefficient –

z Vertical axis m

0 Location at the inner side of the boundary m

0− Location at the outer side of the boundary m

T(r) Surface of triangle prism finite volume m2

kbio Oxygen decay due to biological consumption 1/s

New Unstructured Mesh Water Quality Model 15

Table 6 Symbols for the finitevolume equation and approxi-mated boundary conditions

Symbol Description Unit

ciht ∊ ch

t Concentration vector for a layer h at time level t containing values for allcells listed in I

mol/m3

I Vector list of active cells in a layer –

h Layer index –

s = [1 2 3] Side of a triangle –

t Time level –

Δ Time-wise increment, time-step s

nl(I, s) Neighbor list returning the index for the cell which neighbors to the cells Iat the side s

–

vtih 2 vth Finite volume vector for a layer h at time level t containing values for all cellslisted in I (the cell volume is dynamic due to varying surface elevation)

m3

gihs ∊ ghs ghs ¼ Δathsdths

Δrsvth1/s

Δris 2 Δrs Inter-triangle-center distance to side s m

Δatihs 2 Δaths Surface area of one surface of the finite volume m2

dtihs 2 δths Dispersion coefficient vector at the center of the surface at side s m2/s�vtihs 2 �vths Volume flow m3/s

ɛihs ∊ ɛhs Logical control bit array determining the ratio between explicit and implicitapproximation

–

πihs ∊ πhs Adaptive upwind control bit array –

ktih 2 kth Kinetic constant vector containing individual constants for all cells I 1/s

cat Equilibrium concentration in air mol/m3

keq Henry coefficient –

mtih Measured value mol/m3

stih Source term mol/s

Table 7 Further utilizedsymbols Symbol Description Unit

Y Solution distribution mol/m3

M Matrix the coefficients of concentrations at subsequent time level X

b Vector containing terms approximated at preceding time level andindependent boundary conditions

X

kpxH;inv Inverted Henry coefficient, volatility instead of solubility, as ratioof partial pressure and mass fraction

atm

pg Partial pressure atm

xa Mass fraction –

ρH2O Density of water kg/l

kH Henry coefficient at ratio of molar mass and partial pressure M/atm

ΔsolvH Enthalpy of solution l atm mol−1

R Ideal gas constant l atm mol−1 K−1

a Temperature coefficient K

b Salinity coefficient mol/l

c Constant –

T and T* Temperature and reference temperature K

csalin Salinity, salt concentration mol/l

16 J. Lawen et al.

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New Unstructured Mesh Water Quality Model 17