Mathematical modeling of water quality in river systems. Case study: Jajrood river in Tehran - Iran

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1 European Water 7/8: 3-, E.W. Publcatons Mathematcal modelng of water qualty n rver systems. Case study: Jajrood rver n Tehran - Iran S.A. Mrbagher, M. Abaspour and K.H. Zaman 3 Department of Cvl Engneerng, Khajeh Nasr Toos Unversty, Tehran, Iran. Department of Mechancal Engneerng, Sharf Unversty, Tehran, Iran. 3 Scence and Research Branch Islamc Azad Unversty, Tehran- Iran. Abstract: Keywords: The present research work descrbes the mathematcal model for the Jajrood Rver upstream of Latyan Dam. The rver stretch studed s 5 Km long extendng from Shemshak vllage (upstream) to Latyan Dam (downstream). The Latyan Dam s one of the man sources of water for Tehran metropoltan regon. The Latyan Dam supples 30% of the drnkng water consumed by the ctzens of Tehran. Due to the sewage of resdental areas whch s dumped at the rver and the probable contamnaton, the qualty of water along the rver requres to be tested and the results of the qualtatve analyss should be determned. Water qualty parameters such as Dssolved Oxygen (DO), Bochemcal Oxygen Demand (BOD 5 ), Phosphate Ion (PO ), Ntrte Ion ( ), Ntrate Ion ( 3 ),Ammona Ion( ), Organc Ntrogen (RG), Temperature (T) were tested and modelled n the Jajrood Rver n north of Tehran provnce. A one - dmensonal mathematcal model wrtten by C#008 programmng language wth fnte dfference method and model can be computed hydrodynamc and water qualty Parameters n Jajrood Rver for dfferent reaches. Ths novel model, whch s desgned based on a techncally robust and hghly accurate graphc software, does qualtatve calculatons at a very short tme. It also gves all the nformaton about the occurrences durng the dumpng of sewage and even determnes the hghest concentraton of sewage allowed to be dumped at the rver. The model was calbrated and verfed by usng qualtatve data collected from several statons along the study area form June 006 to June 007. The model was tested durng dfferent months of the year wth satsfactory results. The smulated results of the model are n good agreement wth measured values. The model can be useful n gudng engneerng and management decson concerned wth the effcent utlzaton of Jajrood Rver water and the protecton of ther qualty. water qualty parameters, fnte dfference, pollutant, concentraton, rver systems.. INTRODUCTION Latyan s catchment has 698 km area and 30 m average heght. The stretch of the Jajrood Rver extends from Shemshak vllage (upstream) to Latyan Dam (downstream). The structural features of the dam are as follows: Type: concrete buttress, Heght from the foundaton: 07 meters Gross capacty: 95 Mm 3 Effcent capacty: 85 Mm 3 The Latyan s catchment s vast and lacks meteorologcal statons. Therefore, the data collected from the 6 neghbourng meteorologcal statons were used for hydrologcal and meteorologcal studes. The regresson equatons of the temperature and precptaton gradent are gven below: T= Z, r = 0.95, r = 0.975, n = 6, Tσ n =.77 () R = Z, r = 0.865, r = 0.93, n = 6, Rσ n = 37.8 () T= annual mean temperature ( o C)

2 3 S.A. Mrbagher et al. R= annual mean precptaton (mm) The mean precptaton vared between 50 and 550 mm n Latyan s catchment. Wthn the catchment, ctes and 36 vllages are located. Now, raw sewage of these centers s drectly dscharged about 30 L/s to Jajrood Rver. Dscharge of pollutants from domestc sewage, storm water, and other sources to Jajrood Rver, all of whch may be untreated, can have sgnfcant effects of both short term and long term duraton on the qualty of rver systems. Jajrood Rver s the bggest rver and starts at Shemshak vllage and ends at Latyan Dam. The rver stretch was studed n 5 Klometres. The feld work was conducted from June 006 to June 007. The dscharge amount of Jajrood Rver for that one-year perod vared from m3/s.water samples were analyzed bmonthly from statons specfed before. The locatons of the statons were consdered n the upstream and downstream of the resdental areas whch ther sewage s dumped at the rver. The qualty of Jajrood rver can be greatly nfluenced by the pont and non-pont contamnatng sources n the study area. Generally, the rate of growth of algae can be accelerated and t affects the qualty of water, when the concentraton of Nutrents such as PO and 3 ncreases n the rver water. A quanttatve methodology s requred to estmate the quantty of nutrents, suspended solds, bochemcal oxygen demand and fecal col-form n rver systems and to predct the effects of expected future water qualty. Ths methodology ncludes the development of water qualty parameters transport model n the water body of nterest. A dynamc smulaton approach can be developed on the bass of conservaton of mass to descrbe the effect of BOD 5, DO, PO,, 3,, RG, SS n the rver systems. Suspended solds are the most mportant parameter among others that affect the qualty of water, but t s not the major water pollutant. Although excellent references are avalable n the dscussons of water qualty modellng n stream systems, t s known less about water qualty parameters modellng n rver systems n urban area. Many of water qualty models that exst are theoretcal research models, most of whch have not been appled to real world systems as the requred nput parameters. Model coeffcents are seldom measured n the feld and thus computed results are dffcult to verfy. Several studes were revewed on the ssue of water qualty models. DOSAG s a model whch was developed to solve the ntegrated form of the Streeter-Phelps equatons and s applcable to the system whch can be smulated as one-dmensonal and steady state wth no dsperson. The orgnal DOSAG was modfed for the U.S. Envronmental Protecton Agency (EPA) and the resultng product was called DOSAG III. The QUALII model was an extenson of the stream water qualty model QUALI developed by F.D Masch and Assocates and The Texas Water Development Board (97). In 97, water resources engneers, under contract to the U.S. Envronmental Protecton Agency, modfed and extended QUALI to produce the frst verson of QUALII. Over the next 3 years, several dfferent versons of the model evolved n response to specfc users needs. QUALII s perhaps the most comprehensve n a group of water qualty models developed n the U.S. QUALII s wrtten n FORTRAN77 and very dffcult for users. The majorty of the models are solved by the fnte dfference technque. Major dfferences between the varous models are the number of dmensons consdered and the terms to be retaned n the sources and snks. Fnte dfference models (FDM) are usually restrcted to rectangular grds and, at most, only grd spacng vares. The fnte dfference model s easy to apply to onedmensonal flow problems, but the fnte element model (FEM) has been proved to process the great potental n solvng two-dmensonal flow problems. There have been other studes on water flow and water qualty models usng numercal FEM technques, such as Chang (970), Mrbagher (98). Varable or process n model applcaton s too complex for formulaton and msapplcaton of model calbraton. To overcome these defcences, the new model s wrtten n C#008 wth fnte dfference method that user can work very easly wth model wthout lmtatons. Ths novel model, whch s desgned based on a techncally robust and hghly accurate graphc software, does

3 European Water 7/8 (009) 33 qualtatve calculatons at a very short tme. It also gves all the nformaton about the occurrences durng the dumpng of sewage and even determnes the hghest concentraton of sewage allowed to be dumped at the rver. Wth all the basc data collected from other catchments, ths model can also be appled to those n order to predct water qualty parameters wth a slght change.. MATERIAL AND METHODS. Model formulatons Model formulaton s based on the mass balance for a partcular substance. The statement of the mass balance can be gven as: Accumulaton = nflow outflow ± sources or snks. The mathematcal model conssts of one-dmensonal advecton-dsperson mass balance equaton of the gven polluton parameter, correspondng, ntal and boundary condtons. The assumptons n such model are:. The densty of polluted water s constant and smlar to clean water densty.. The substance s well mxed over the cross secton. (The assumpton has been proved based on the measurements done.) 3. Only longtudnal hydrodynamc dsperson occurs. The one-dmensonal equaton for the conservaton of mass of a substance n soluton,.e. the one dmensonal advecton-dsperson equaton for the concentraton of Dssolved Oxygen (DO), Ntrate ( 3 ), Ntrte( ), Ammona ( ), Orthophosphate (PO ), Bochemcal Oxygen Demand (BOD 5 ), Organc Ntrogen (RG) and Temperature (T) can be wrtten for a rver flow systems assumng steady-state, non-unform, as follows: (. x. ) ( A. Dx. c c AV C ) χ dc S + = + + { t 3 A x A x3 dt 3V storage advecton dsperson source & snks (3) C=concentraton of pollutant (mg/l) T=tme (s) X=dstance A=cross-sectonal area perpendcular to x (m ) D x =dsperson coeffcent (m /s) V x =water velocty perpendcular to x (m/s) S=source or snk pollutant (mg/s) V=volume (L) Dsperson coeffcent was determned by: D x =.0 n V x D () D x : Longtudnal dsperson coeffcent (m /s) n: Mannng s roughness coeffcent V x : mean velocty (m/s)

4 3 S.A. Mrbagher et al. D: mean depth Sources and snks for the model n the unt of volume are as follows:. For Bochemcal Oxygen Demand: S = k L k 3 L (5). For Dssolved Oxygen: S K L K o K K + K O s + K A (A + A) = (6) 3. For Ammona: S = K + K Kg RG (7). For Ntrte: S = K K (8) 5. For Ntrate: S = K K B ( A + A ) (9) 6. For Organc-N : S = K kg RG (0) 7. For Orthophosphate: S = K PO PO K PO ( A + A ) () K = Boxdaton coeffcent for BOD (/day) K 3 = Coeffcent of settlng effects (/day) L = BOD Concentratons (mg/l) K = Re-aeraton Coeffcent (/day) O s = Oxygen saturaton pont (mg/l) O = DO Concentraton (mg/l) K A = DO from chlorophyll-a (/day) A = Chlorophyll- A Phytoplankton (mg/l) A = Chlorophyll-A sessle algae (mg/l) K = Ntrte or ntrate rate (/day) 3 = Ntrate concentraton (mg/l) K B =Ntrate uptake by algae (/day) = Ntrte concentraton (mg/l) K = Ammona decay rate (/day) RG =Organc N concentraton (mg/l) = Ammona concentraton (mg/l)

5 European Water 7/8 (009) 35 K kg = Organc-N decay rate (/day) K PO = Orthophosphate decay rate (/day) K PO = Orthophosphate uptake by algae (/day) PO = Orthophosphate concentraton (mg/l). Model conceptualzaton The model conceptualzes the stretch of rver studed n statons specfed before ( reaches). The parameters of BOD 5, DO, Ntrfcaton, Orthophosphate, and temperature have been measured for the perod of one year. Reaches are further subdvded nto unts called computatonal elements. Each computatonal element s modelled as a constant volume, completely mxed reactor wth nput, output, and reacton terms. The mathematcal model contans subroutne coverng the qualty parameters, determnaton of coeffcents, temperature calbraton and hydraulc subroutne. These subroutnes nteract together wth the man model to smulate the water qualty parameters of the Jajrood Rver. Generally, these subroutnes smulate each qualty parameters by defnng the nature of the source/snk, S, n the mass balance equaton. The element conssts of two parts: The frst contans the forcng functon nputs whch enter drectly nto the mass balance equaton. The second conssts of the reactve components whch are specfc to the partcular water qualty varable beng modeled. The hydraulc parameters were calculated va the Mannng s Equaton: Q= /n A R 0.67 S 0.5 () Q: flow rate (m 3 /s) n: roughness A: cross secton area (m ) R: Hydraulc radus S: Slope.3 Calbraton Coeffcent In the model, the coeffcents of BOD, DO, Ntrfcaton, Orthophosphate are calculated and calbrated accordng to the stream temperature by the general formulas: k 0 = (log X -log X )/(t -t ) (3) k T =k 0 θ(t-0) () X= concentraton of consttuents (mg/l) t: tme (day) k T = chemcal reacton coeffcent at temperature T (/day) k 0 = chemcal reacton coeffcent at 0 C (/day) θ= temperature correcton factor. K 0 s calculated by the measured data for the dstance between the two statons and s compared wth the data n the reference books. Then, consderng the measured temperature, the coeffcents are calbrated.

6 36 S.A. Mrbagher et al.. Soluton scheme In the present study, an mplct fnte dfference scheme s used for the numercal soluton of the advecton-dffuson equaton. In ths method the fnte dfference approxmaton expresses the values and the partal dervatve of each functon wthn a four pont grd formed by the ntersectons of the space lne -, and + wth the tme lnes t n and t n+. A control volume s defned and stuated around the grd pont. The boundares of ths control volume are rver bed, the water surface and the two cross-sectons stuated at - and +, respectvely, as shown n Fgure. Fgure. Classcal Implct Nodal (Obtaned from QualE Manual) For a dscrete tme nterval Δt, begnnng at t n and collectng term n concentraton (C), the resultng fnte dfference form of Equaton (3) s: AD Δt + AD AD Δt Q + ( x )..C n Q + ( x ) + ( x ) + AD Δt Δx v C n + = Z Δx Δx v ( x ) C n v + Δx (5) z n = C + S Δt + ΔtQ x.c x (6) v = ( A Δx + A. Δx ) (7) t x / Vx (8) By usng Equaton (7) n Equaton (5) and assgnng the letters α, β and γ for all terms on the left-hand sde of Equaton (5), the followng coeffcent results for tme step n+: α Δt = ( D X ) Δx Δt Q - v [(D x ) + (D x ) ] β Δt Δt = + + Q Δx v (9) (0)

7 European Water 7/8 (009) 37 = ( D x ) Δt Δx () Usng Equaton (9) through () n Equaton (5), yelds the algebrac equaton: α C n+ C n C n β δ + = z () n The term Z n equaton (6) vares dependng on the consttuent beng consdered. Ths term s, as t s defned n equaton (6), takes nto account the source and the snk. At each tme step, Equaton () s wrtten once for each computatonal element,, n+; therefore, I N, whch results n a system of N smultaneous equatons and N unknown..5 Intal and boundary condton If the concentratons at frst staton n upstream rver are known at the begnnng of the smulaton perod, they can be used as ntal condtons, and user can calculate the concentraton of consttuents n the other applcaton. Snce the rato Δ x / Δt n the numercal scheme should be approxmately equal to the current velocty of water n the prototype, consttuents wll travel a dstance x n a tme nterval t. Thus f, only advecton operates the downstream boundary condton can be gven approxmately by n + n+ c c, where c s the concentraton just downstream from the end of the system. = n Model applcaton The model was appled to smulate Dssolved Oxygen, Bochemcal Oxygen Demand, Ntrte, Ntrate, Orthophosphate, Ammona, Organc Ntrogen and Temperature n the Jajrood Rver. Jajrood Rver s located between Shemshak vllage and Lavasan cty. The length of the rver s 5 Klometers. The peak flow of the Jajrood Rver was measured 30. m 3 /s for the perod of one year and the average precptaton n Latyan dam s catchment s 500mm per year. Jajrood Rver was dvded nto eleven reaches and twelve samplng stes for model calbraton and valdaton. The total prmary staton s nne. The number of statons, rate of flow, velocty, cross-secton area, and the sze of the grd are shown n Tables -3 for 3 months. The water qualty tests were performed from June 006 to June 007. Each reach was then dvded nto several computaton elements, havng ther own hydraulc, physcal, chemcal and bologcal characterstcs. The nput data for model valdaton are topographcal and hydraulc data and water qualty n the samplng ste. Topographcal data are rver cross-secton measured at all samplng stes. Requred hydraulc data were: flow rates, water depth and velocty. They were measured monthly n one year. Rver branches between the statons were nput to the model as ncremental flow. The temperature of the stream durng the survey perod was measured drectly n the feld. The man data requred for ths model are as follows: Number of major statons Geometrc and hydraulc characterstcs of the statons Qualtatve parameters measured at the frst staton Length between the major statons Number of the branches leadng to the man rver wth the geometrc and hydraulc characterstcs Dstance to the other statons Temperature at the statons

8 38 S.A. Mrbagher et al. Locaton of the ntersecton pont of the sewage and the man rver wth hydraulc and qualtatve characterstcs Roughness coeffcent for the length of the rver The hydraulc, hydrodynamc, qualtatve coeffcents and the concentraton are calculated and smulated n the model. Also the model draws and compares the profles of the smulated and measured concentraton of the consttuents n a dagram. Table. Number of prmary staton, Rate of flow, Velocty, Cross secton area and the Sze of the grd (Nov. 006) Prmary Staton Elevaton Length Cross secton area (m ) Veloct y(m/s) Rata of flow (m 3 /s) Number of dvson between two staton Sze of grd Table. Number of prmary staton, Rate of flow, Velocty, Cross secton area and the Sze of the grd (Feb. 007) Prmary Staton Elevaton Length Cross secton area (m ) Veloct y(m/s) Rata of flow (m 3 /s) Number of dvson between two staton Sze of grd Table 3. Number of prmary staton, Rate of flow, Velocty, Cross secton area and the Sze of the grd (Apr. 007) Prmary Staton Elevaton Length Cross secton area (m ) Veloct y(m/s) Rata of flow (m 3 /s) Number of dvson between two staton Sze of grd RESULTS AND DISCUSSION Functonal relatons for suspended solds, flow rate, total ntrogen, and ntrate were also obtaned through monthly samplng n the last staton from June 006 to June 007 wth regresson methods. The number of samplngs was 5 for the afore-mentoned one year perod.

9 European Water 7/8 (009) 39 The functonal relatons are: N TOT = Ln Q, r =0.76, r = 0.87, n=5,nσ n =0., _ N =0.33 (3) SS=-0.0Q +0.7Q 3 -.5Q Q-38.88, r = 0.90, r=0.95, n=5, SSσ n =6, N TOT = , r =0.988, r=0.99, n=5, Nσ n =0., Usng Eq. (5) n Eq. (3), yelds the anther Equaton: SS =03 () _ N =0.33 (5) 3 = Ln Q (6) N TOT = concentraton of total ntrogen (mg/l) Q= flow rate (m 3 /s) SS= concentraton of suspended solds (mg/l) 3 =concentraton of ntrate (mg/l) The water qualty parameters such as BOD 5, DO,,, 3, RG, PO are gven to the model after conductng the samplng test n the frst staton and then, the same parameters are smulated for other 8 statons by the model. The results of the model are shown and compared wth the measured values n Tables -6 from 9 prmary statons. The above-mentoned tables show that smulated values of water qualty parameters have acceptable compatblty wth the measured values and the dscrepances between the measured and smulated are small. Therefore, the applcaton of ths model was proved for calculatng water qualty parameters.. CONCLUSION A one-dmensonal dynamc mathematcal model calculates Dssolved Oxygen, Bochemcal Oxygen Demands, Ntrate, Ntrte, Orthophosphate, Ammona, Organc Ntrogen, and Temperature n rver systems. Ths model s based on the mass balance equaton ncludng advecton and dffuson transport, chemcal and bologcal transformaton and nflow of pont and non-pont sources. The soluton s obtaned by a fnte dfference mplct method. The model was calbrated and verfed by usng qualtatve data collected from several statons along the study area form June 006 to June 007. The obtaned data were measured wth great accuracy based on whch the functonal relatons were also created between (N TOT and Q), (SS and Q), and (N TOT and 3 ) wth acceptable regresson coeffcent. The model was tested durng dfferent months of the year wth satsfactory results. The smulated results of the model are n good agreement wth measured values. The model can be useful n gudng engneerng and management decson concerned wth the effcent utlzaton of Jajrood Rver water and the protecton of ther qualty. The smulated results have acceptable compatblty wth the measured values. The advantages of ths model are as follows:

10 0 S.A. Mrbagher et al.. Beng appled easly wth techncally robust, hghly accurate computer software wth hgh speed. Drawng the profle of water qualty parameters along the rver 3. Calculatng the water qualty parameters. Determnng crtcal concentratons 5. Plannng to control the pollutants 6. Applyng to other rvers wth the smallest changes n the structure of the model 7. Beng updated easly n the future Table. Comparson of measurement amount of water qualty of Jajrood rver wth the smulated result of model (Nov. 006) Prmary BOD 5 (mg/l) DO(mg/L) (mg/l) (mg/l) 3 (mg/l) RG(mg/L) PO (mg/l) Staton M * S ** M S M S M S M S M S M S * Measured ** Smulated Table 5. Comparson of measurement amount of water qualty of Jajrood rver wth the smulated result of model (Feb. 007) Prmary BOD 5 (mg/l) DO(mg/L) (mg/l) (mg/l) 3 (mg/l) RG(mg/L) PO (mg/l) Staton M * S ** M S M S M S M S M S M S * Measured ** Smulated Table 6. Comparson of measurement amount of water qualty of Jajrood rver wth the smulated result of model (Apr. 007) Prmary BOD 5 (mg/l) DO(mg/L) (mg/l) (mg/l) 3 (mg/l) RG(mg/L) PO (mg/l) Staton M * S ** M S M S M S M S M S M S * Measured ** Smulated

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