MATHEMATICAL DETERMINATION OF PARAMETERS OF DISSOLVED OXYGEN SAG EQUATION

Abstract

When sewage is added into a body of water, the receiv-ing water annuls the pollution by the combined action of natural forces of purification such as dilution, sedimenta-tion, oxidation, reduction, sunlight, reaeration, etc. The principle of self- purification of polluted water is utilised in the method of disposal of sewage by dilution. A polluted river purifies itself under four zones viz0 zone of degration, active decomposition,, recovery and cleaner water. The interplay of forces of deoxlygenation (due to BOD)C of reaeration (from the atmosphere) in the polluted water on creates a spoon-shaped profile of DO deficit along the path of water movement. This profile is called the DO Sag Curve. Streeter & Phelps were the first to give the following mathematical relationship expressing DO deficit at various times in a stream. D = kLa (ektrt) ~ Da ert r-k This equation has the usual notations. Since its inception, this equation has always attracted lot of research and application. So many modifications of the Streeter & Phelps equation were proposed. Towards its application, one of the problems has been to find values of the constant parameters r,k and La. While k & La could be easily found by conducing BOD test in the laboratory, r could not be found easily and reliably. iv Pew approaches were given out by different workers for finding value of r, Notable amongst them were the works of streeter, O'Connor, Dobbins etc. who have chosen to express r as a function of hydrologic parameters of the river such as water depth, velocity, flowing section etc. But such formulae could not give values of r satisfying the field observations of D.O. concentration values. This difference has been argued to be due to action of algal cells, sedimentation and other factors in the river whichw were not directly considered when S&P Equation was applied. But it has been perhaps only a fallacy of thinking since effects of all factors involved in self-puri-fication process in the river will be manifested into the resultant value of ro Thus there is a need to improve the method of finding correct values of r and other parameters of S&P Equation. An integrated approach for finding r,k.and La as may be Prevalent in the river? has therefore been proposed in the present study where only DO - time data will be used to compute values_of these parameters. In the present study this method has been applied to the observed. DO - time data of James river USA taken from the paper of D.J. O°Connor, "Oxygen Balance of an Estuary" of Journal Sanitary Engineering Division, ASCE, SA3 May 1960, page 73. ® v v In the proposed method, theoretical D &~Dt. values have been computed from the Streeter & Phelps Equation and a xx common set of values of r and k has been found with the help of graph satisfying or matching the observed values of.D and Dt. To ascertain the goodness of the fit, the observed and calculated, values of DO Concentiation were plotted against time. The results were found to be quite matching. The values of f calculated were also compared with the rough values worked out by Mull's method. Thus the proposed method of determination of in-situ or in-river values of parameters of S&P Eqn. Viz. r,k & La has been found to be quite suitable. The proposed method would avoid the errors in determination of r and k under imposed conditions of laboratory/field and assumed conditions of analysis. The proposed method needs and uses only DO concentration versus time data along the river. Weightage of all the data points is considered in the proposed method. Under the further work it has been suggested that the values of D and Dt should be computed for wider and more closer values of r and k to cover all cases of sag curve analysis, and that more work of similar type be carried out for a number of DO sag curves® Further the values of the parameters so found be compared with the field values.