FORECASTING DISSOLVED OXYGEN AND BIOCHEMICAL OXYGEN DEMAND IN A RIVER WITH NUMERICAL SOLUTION OF ONE-DIMENSIONAL BOD-DO MODEL

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1 - - FORECASTING DISSOLVED OXYGEN AND BIOCHEMICAL OXYGEN DEMAND IN A RIVER WITH NUMERICAL SOLUTION OF ONE-DIMENSIONAL BOD-DO MODEL WANG, S. H. LI, R.,,* School of Economics and Management, Beihang University, Beijing 9, China Bsiness School, Beijing Normal University, Beijing 875, China *Corresponding athor liri67@bn.ed.cn (Received 4 th Oct 6; accepted th Dec 6) Abstract. As a main measre of water qality, the concentration of DO always gets mch attention. The chemical reactions in water bodies are mainly related to DO, so the concentration of DO has mch relationship with the water qality. Biochemical oxygen demand and dissolved oxygen (BOD-DO) model is sed to show the relationship of the concentration of BOD or DO with the physical characteristic of the river. For this model, it is difficlt to determine the analytical soltion, so nmerical soltion of the model is obtained with Chebyshev orthogonal polynomial. With this method, the model is rewritten with differential form, and then the for order differential of the oxygen deficit was expressed with the Chebyshev orthogonal polynomial, which had coefficients. Lastly, a simlation test was condcted to verify the rationality of the model. The actal BOD and oxygen deficit vales are calclated sing the original model. With the actal vales and formlas, the coefficients cold be solved and the predicted vales calclated. The forecasting vales of concentration of BOD and DO are compared with the actal vales. Five statistical measres were sed to evalate the predicted reslts. Keywords: water qality; analytical soltion; Chebyshev orthogonal polynomial; statistic index; oxygen deficit Introdction Many water bodies have become pollted de to rapid economic and social development (Q and Fan, ; Zhang et al., ; Li et al., ; Ma et al., 9). River water qality has been sbstantially affected by indstrial, agricltral and mnicipal waste water. So it is necessary to assess and predict water qality (Sn et al., 5). The main polltion indicators are biochemical oxygen demand (BOD), five day biochemical oxygen demand (BOD 5 ) and the permanganate index (COD Mn ). From China official report, among 6 lakes and reservoirs in China, 5.% are etrophic, 8.% are mildly etrophic and 6.7% are moderately etrophic. The main indicators are total phosphors (TP), five-day biochemical oxygen demand (BOD 5 ) and the permanganate index. In the same time, the concentration of dissolved oxygen (DO) is always low which makes the polltion more serios (Ministry of Environmental Protection of the People s Repblic of China, ). Water qality assessment evalates the polltion levels of rivers or water areas with qality or qantity indicators (Y et al., 6; Chen et al., 5). APPLIED ECOLOGY AND ENVIRONMENTAL RESEARCH 5():-. ISSN (Print) ISSN (Online) DOI: 7, ALÖKI Kft., Bdapest, Hngary

2 - 4 - Water qality mathematical models can help establish the relationship between the emission of polltants and changes in water qality. Sch models change a complicated river system into sitable mathematical eqations and simlations (He et al., ; Lin et al., 5). As comprehensive indicators that reflect the organic polltion of water body, biochemical oxygen demand (BOD) and dissolved oxygen (DO) are two important parameters for jdging the degree of water cleanliness. The well-established BOD-DO water qality model describes the change the law of BOD and DO in a river and is a matre water qality model. It forms the basis of many amended and complex BOD-DO models (Yan et al., ; Zeng et al., ). The analytical soltion of the model is difficlt to obtain and it is necessary to determine the nmerical soltion (Zh et al., ). Revelli and Ridolfi established a one-dimensional water qality model and obtained the probability density fnction of BOD (Revelli and Ridolfi, 4). The finite element and Monte Carlo methods were also sed to solve the water qality model separately (X et al., 4a; Go et al., 4), and the two methods were combined to solve the problem (X et al., 4b). STREAM II modeling package simlates the DO and BOD parameters in a two-dimensional model (Sharma and Singh, 9). Materials and methods BOD-DO model eqations The research and development of water qality model has gone throgh a nmber of stages (Fan and Lv, 8; F, 987). The first stage was from 95 to 98, dring which the research object of water qality models was the water body, inclding the components of water qality. In 95, Streeter and Phelps proposed the first water qality model. Based on their research, other scholars sed and improved the model for water qality forecasting (Go et al., ). The second stage was from 98 to 995, involved the se of water qality models in more complicated systems and in combination with the watershed models, ths allowing non-point polltion sorces to be treated as initial inpts (Liao and Tim, 997). The third stage was from 995 to the present dring which the water qality models were developed into a comprehensive model from reaction models. The most common research interests have concentrated on etrophication of reservoirs and lakes (Qan and Yan, ). The one-dimensional reach is the minimm nit of a river. In this sitation, there is only one sewage otlet or tribtary at beginning of the reach. The BOD-DO model describes the changes in polltant concentration, and is expressed by oxygen deficit as: L L Dx k L x x D D Dx k L kd x x (Eq.) APPLIED ECOLOGY AND ENVIRONMENTAL RESEARCH 5():-. ISSN (Print) ISSN (Online) DOI: 7, ALÖKI Kft., Bdapest, Hngary

3 - 5 - where L is the concentrations (g/m ) of BOD, is the flow velocity (km/day or km/d) of water, D is the oxygen deficit, (g/m ), D x is the dispersion coefficient (m /d), k is the attenation coefficient of BOD (d - ), and k is the coefficient of reoxygenation of river (d - ). Under steady state, the concentration of indicators does not change, that is L C =, = t t. When considering the effect of dispersion, the analytical soltion of the eqations can be expressed as follows: x L= Le kx/ kl kx / kx/ D De ( e e ) k k (Eq.) The expressions of and are both complex formlas and there is no reglar patterns to obtain their approximate vale. So the two eqations can be transformed as follows: 4Dxk 4Dxk ( ) k = = Dx 4Dxk 4Dxk 4Dxk 4Dxk ( ) k D 4D k 4D k x x x (Eq.) From the transformation above, we can conclde that when the parameters obey the relationship D k D k and, the model has an analytical soltion. For other x x sitations, however, it is difficlt to obtain the analytical soltion for the model. BOD-DO model soltion As referred to above, the BOD-DO model determines the relationship between the concentration of BOD and DO and the river location. In Eq. (), some parameters need to be estimated. First, the nmerical soltion is obtained with the parameters, and then certain methods are applied to obtain the parameter vales. The next step shows the processes to solve the BOD-DO model. From the first part of Eq. (), we can obtain the expression of L with other parameters and symbols. The expression of L, the derivative of L, and the second derivative of L are smmed p as follows: APPLIED ECOLOGY AND ENVIRONMENTAL RESEARCH 5():-. ISSN (Print) ISSN (Online) DOI: 7, ALÖKI Kft., Bdapest, Hngary

4 - 6 - d D dd L= ( Dx k D) k dx dx dl d D d D dd = ( Dx k ) dx k dx dx dx 4 d L d D d D d D = ( D x k 4 ) dx k dx dx dx (Eq.4) After the calclation, the three eqations above can be sbstitted into the second formla of Eq. (). Eq. (4) shows the relationship of the differential and the parameters of the model. We can conclde that the soltion of the model refers to a high order-nonlinear problem. The Chebyshev orthogonal polynomial is an effective way to solve this high order differential eqation. D d D d D k d D k dd 4 x D 4 x ( D ) ( ) x Dx k D (Eq.5) k dx k dx k k dx k dx Chebyshev orthogonal polynomial When the interval is [-, ], and the weight fnction is ( x), the orthogonal polynomial combined with the orthogonalization series {, x,, x n, } is called the Chebyshev orthogonal polynomial. The expression of the nth item is Tn ( x) cos( narccos x), x. Let x cos, when x varies in the interval [,] and varies in the interval [, ]. The expression can be rewritten as Tn ( x) cos( n ), instead. The general term formla is n [ ] n ( n m )! Tn ( x) ( ) ( x) m!( n m)! m m nm x, with Tn(x) being the polynomial of degree n with the coefficient of the first item being n-. The recrsion formla of the differential coefficient form of the Chebyshev orthogonal polynomial is d d Tn ( x) Tn ( x) Tn ( x) n dx n dx. The vale of x is between - and, which is not appropriate for river position. Notation z can be sed to replace x, sch that z a( x), with the interval of z being from to a. The relationship of x and z is given as follows: z dx x,, a. a a dx According to the relationship of x and z, the connection of and z can be dedced. The Chebyshev orthogonal polynomial can be expressed with z. T n (z) = n [ n ] å (Eq.6) m= (n- m-)! (-) m m!(n- m)! [(- z a )]n-m APPLIED ECOLOGY AND ENVIRONMENTAL RESEARCH 5():-. ISSN (Print) ISSN (Online) DOI: 7, ALÖKI Kft., Bdapest, Hngary

5 - 7 - When z a( x), the recrsion formla of the differential coefficient form of the Chebyshev orthogonal polynomial is changed into: dtn ( z) dtn ( z) Tn ( z) n dx n dx (Eq.7) Solving the BOD-DO model with the Chebyshev orthogonal polynomial To solve the problem mentioned in Eq. (5), the for order derivate of the concentration of the dissolved oxygen deficit, Chebyshev orthogonal polynomial. 4 d D 4, shold be expanded by the (Eq.8) The next step is to integrate 4 d D 4. When performing integration, the vale of T d Hf is lost and it shold be added when calclating z z dd. When z z, the vale T of d Hf z z is a constant, denoted as A. Becase f consists of the polynomial of z, d T z f is still composed of the polynomial of z. We can then obtain the expression of z dd in Eq. (9). The calclation of calclation of dd d D, dd and D are mch the same as the. Eq. (9) shows the expressions of these derivatives. Like A, the vales of A, A and A are all constants. d D = dt Hf + A, d D = dt H f + A, dd = dt H f + A, D = d T H 4 f + A (Eq.9) The five eqations from Eq. (8) to Eq. (9) are sbstitted into the first part of Eq. () and the expression of L can be expressed with the Chebyshev orthogonal polynomial. Lastly, the derivatives of D to x are sbstitted into Eq. (5). The problem of obtaining the soltion of the BOD-DO model is now trned into solving Eq. (). APPLIED ECOLOGY AND ENVIRONMENTAL RESEARCH 5():-. ISSN (Print) ISSN (Online) DOI: 7, ALÖKI Kft., Bdapest, Hngary

6 - 8 - D k ( ) ( )( ) k K k k x T T T d f Dx d Hf A Dx Dx d H f A k d H f A k d H f A T T 4 ( )( ) ( ) k (Eq.) Becase other parameters can be obtained or calclated with chemical or physical indicators, the main step dring the process of solving Eq. () is to solve matrix H. Matrix dd z d f d Hf d Hf A T T T From Eq. (9), zz +, the expression of H can be z dedced as z z f Hf. To obtain the expression of matrix H, the first step is from the a d a d derivate form: Tn ( z) Tn ( z) Tn ( z), in the interval [ z, z ]. If t, ( n ) ( n ) then z(), that is to say z. When n and n, performing integration a d a d T ( z) T ( z) T ( z) ( n ) ( n ) for the eqation: n n n. In water qality analysis, z means the position is the start of the river. In this sitation, a d a d T ( z) T ( z) T ( z) ( n ) ( n ) z z z can replace this formla and the z n z n n z formla can be simplified (Eq. ()). The vale of the two expressions can then be a ( n ) n z determined, they are T ( z) below: a ( n ) n z and T ( z). The analysis is shown a d a d T ( z) T ( z) T ( z) ( n ) ( n ) z z z n n n a a a T ( z) T ( z) T ( a) T ( z) ( n ) ( n ) ( n ) ( n ) n n n z n z (Eq.) When n =,. Ths, the first row of matrix H has jst two elements that are not zero,. When n =,. From this we can obtain the second row of matrix H, which also has two elements that are not zero,. These two rows can only be observed according to the form z T ( z ) APPLIED ECOLOGY AND ENVIRONMENTAL RESEARCH 5():-. ISSN (Print) ISSN (Online) DOI: 7, ALÖKI Kft., Bdapest, Hngary

7 - 9 - z and T ( z ). When n, however, we can obtain the recrsion formla. According to Eq. (), the next problem is to calclate the vale of the last two items of this eqation. a (n+) T (z) - a n+ z= (n-) T (z) n- z= [ n+ ] = a (n- m)! å (-) m n-m - a (n- - m)! å (-) m n-m - 4 m!(n+- m)! 4 m!(n-- m)! m= [ n- ] m= (Eq.) H n When n () a ( n)( n), the vale of the corresponding row of the matrix are H ( n) a ( n ), n, and n H ( n ) a ( n ). According to the analysis above, the strctre of matrix H can be obtained as,,, H n () a, Hn ( n) a ( n)( n ) ( n ), and n H ( n ) a ( n ). Reslts and discssion To illstrate the effectiveness of this method in solving the BOD-DO model, we condcted a simlation experiment to determine if the method cold be sed to predict water qality. According to the empirical data and actal sitation, we assmed that the river was in a steady state and that there were no tribtaries or other sewage otlets in the stdy reach. The parameters of the model mentioned above were also given proper vales, D.48 g / m, 6 km / d, L 64. ml / L, k.4 d, k.8 d, and Dx 8.5 m / d. The distance between two sample sites was km and there were 5 sites in total. Figre. Actal and predicted data of BOD APPLIED ECOLOGY AND ENVIRONMENTAL RESEARCH 5():-. ISSN (Print) ISSN (Online) DOI: 7, ALÖKI Kft., Bdapest, Hngary

8 - - According to the initial eqation (Eq. ), the actal vales of L and D in each site were calclated. Then the coefficient vales of the Chebyshev orthogonal polynomial expansion were obtained by Eq (). Lastly, the predicted vales of L and D were acqired. The actal vales and predicted vales of BOD and DO were compared and the reslts are shown in Fig. and Fig.. The crve of the predicted data was very close to that of the actal data. Althogh in some sites, the predicted data were not very accrate, the forecasted reslts as a whole followed the same trend as the actal sitation. Ths, the soltion method solved the BOD-DO model. Figre. Actal and predicted data of oxygen deficit Comparing the forecast vale with the actal vale on the srface, five statistical assessment indices, specifically R Sqared, the mean absolte error (MAE), the mean absolte percentage error (MAPE), the mean sqared error (MSE) and the root-mean-sqare error (RMSE), were also sed to evalate whether the predicted reslts were good or bad (Table ). Table. Statistical assessment measres for model prediction L (concentration of BOD) D (oxygen deficit) R Sqared MAE.9.89 MAPE.4.58 MSE RMSE APPLIED ECOLOGY AND ENVIRONMENTAL RESEARCH 5():-. ISSN (Print) ISSN (Online) DOI: 7, ALÖKI Kft., Bdapest, Hngary

9 - - The coefficient of determination, denoted as R (R sqared), indicates how well data points fit a statistical model. The R sqared of L and D were.996 and.9775 respectively. These vales were very close to, demonstrating that the forecast vales of L and D cold explain the concentration trends of DO and BOD. The mean absolte error (MAE) is sed to measre how close predictions are to evental vales. From Table, the MAE vale for BOD was.9, which was a little higher, and was.89 for oxygen deficit. However, this measre was an absolte vale not a percentage vale. Frther, the interval of the centration of BOD was from to 64. ( g/ m ) and the interval of oxygen deficit was from to.48 ( g/ m ). This might explain why the MAE vale for BOD was a little larger. The mean absolte percentage error (MAPE) measres the accracy of a method for constrcting fitted time series vales, specifically in trend estimation. It sally expresses accracy as a percentage. This prodces a percentage vale and appropriate for BOD and oxygen deficit. The mean sqared error (MSE) of an estimator is way to qantify the differences between forecast and actal vales of the qantity being estimated. An MSE of zero, meaning that the forecast vale predicts observations of the parameter with perfect accracy, is the ideal, n however, it is practically impossible to achieve. MSE is calclated by MSE = å (F t - A t ) / n. The MSE of the oxygen deficit was less than., which was an ideal vale for the model. Similar to MAE, however, this vale is a mean sqared error and it also magnifies the error. The MSE vale of BOD was.5596 which was a little large, bt still within the scope of acceptability. The root-mean-sqare deviation (RMSD) or root-mean-sqare error (RMSE) is a freqently sed measre of the differences between vales predicted by a model or an estimator and the vales actally observed. RMSD is n a good measre of accracy and is calclated as RMSD = å (F t - A t ) / n. This measre is the sqare root of MSE, so it has a similar trend to MSE. Becase MSE is a sm of sqare vale, RMSD is more objective and reasonable. They were sed to evalate the rationality of the present model, with the reslts (Table ) illstrating that solving the nmerical soltion with the proposed method was very sitable. t= t= Conclsion In this stdy, a new method was proposed to obtain the nmerical soltion of the differential eqation. It was sed to solve the standard BOD-DO water qality model. The derivative of BOD of the model was calclated first. The for orders, three orders, and one order of oxygen deficit were also calclated. The for order differential of oxygen deficit was expressed by the Chebyshev orthogonal polynomial and corresponding coefficients. With the integral of the for orders differential, other differential eqations were also expressed. Ths, the BOD-DO model cold be denoted with the Chebyshev orthogonal polynomial. After that, we solved the coefficients with APPLIED ECOLOGY AND ENVIRONMENTAL RESEARCH 5():-. ISSN (Print) ISSN (Online) DOI: 7, ALÖKI Kft., Bdapest, Hngary

10 - - simlation data. With the soltion method and coefficients, the simlated vales of BOD and oxygen deficit were calclated. Two figres were sed to compare the actal and predicted data. Also there are some irreglar points that deviated from the crve, the reslts as a whole is receivable. Five statistical measres were also sed to evalate the reslts of the model. These reslts showed that the predicted vales were very similar to the actal vales. The simlation test showed that the present method sing the Chebyshev orthogonal polynomial to obtain the nmerical soltion was acceptable and reasonable. Acknowledgements. This work was spported by Yoth Scholars Program of Beijing Normal University (No. 4). REFERENCES [] Cai, Y., Pan, W., Ren, L. (5): Commentary on Basins. System. - Safety & Environmental Engineering (): [] Chen, J., Zh, L., Ma, H. Sn, L. (5): Fzzy clster model based on entropy coefficient assessment of raw water qality - Water Resor. and Power. (6). [] Chen, Y., Zhang, Y., Chyng, F. Y. (9): Modelling techniqes of drainage network in UK. - China Water and Wastewater 5(): 7-. [4] Fan, S., Lv, J. (8): Stdy and progress of water environment nmerical model. - Shanxi Archit 4(): -. [5] F, G. (987): Water qality mathematics model of river and simlation. - China Environ. Sci. Press, Beijing, China. [6] Go, J., Li, S., Long, T. (): Progress of water qality model and its application. - J. Chongqing Jianzh Univ 4(): 9-5. [7] Go, L., Cai, G., Zeng, G. (4): Application of two-dimensional stochastic model of water qality in simlating polltion zone. - J. Cent. Soth Univ (Sci. and Technol.) 5(4): [8] He, J. X., Li, Z. L. (): A review on biofel prodction from biomass by microorganisms. - Jornal of Pre & Applied Microbiology 7(4): -6. [9] Li, S., Li, H., Xia, J. (5): Dapeng bay water environment capacity analysis on the base of Delft D model. - Res. on Environ. Sci 8(5): [] Liao, H., Tim, U.S. (997): An interactive modeling environment for non-point sorce polltion control. - J American Water Resor Assoc (): [] Lin, Z., Peng, W. X., Ge, S. B., Li, D. L., Frta, Y. Z. (5): Strctre Characteristics of Oxidation Pretreated Fiber and Biochemically Binded Boards against Gravida Abortion. - Jornal of Pre and Applied Microbiology 9(): 7-4. [] Li, H., Yang, Y., Kang, J., Fan, M., Q, J. (): Removal of tetracycline from water by Fe-Mnbinary oxide. - J Environ. Sci. 4(): [] Li, W., Li, H., X, H. (9): Calclation method of water environment capacity for water fnction area based on MIKE model. - Water Resorce and Pydropower of Northeast 8: [4] Ma, X., Tang, J., Shen, Y., Fan, M., Tang, H., Radosz, M. (9): Facile Synthesis of Polyester Dendrimers from Seqential Click Copling of Asymmetrical Monomers. - J. American Chem. Soc. (4): [5] Ministry of Environmental Protection of the People s Repblic of China (): Environ. - Bll of China. APPLIED ECOLOGY AND ENVIRONMENTAL RESEARCH 5():-. ISSN (Print) ISSN (Online) DOI: 7, ALÖKI Kft., Bdapest, Hngary

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