MATHEMATICAL MODELLING OF BOD - DO RELATED PROBLEMS IN A STREAM

Abstract

The problem of low dissolved oxygen (DO) begins with the oxygen demanding waste input into a river. The aerobes utilize the river's DO to stabilize the organic waste entering into the river. The amount of oxygen consumed by bacteria to decompose aerobically the organic matter in a specified period of time and at a stated temperature is known as biochemical oxygen demand (BOD). The existing model for DO sag parameters (such as De, the deficit at the critical point and Di, the deficit at the point of inflection occurring at the time to and ti respectively) are not in a very conveniently usable form. Moreover, the determination of De require prior determination of te. The polynomial expressions for all the DO sag parameters are derived in terms of non-dimensional parameters. These polynomial expressions can predict the parametric values within ± (2.5-7.5)% accuracy. The polynomial expressions can replace the Fair's model for DO sag. The nomographs have been prepared. The assumption made by Streeter and Phelps that advection is an exclusive transport mechanism unnecessarily restricts model's validity with the availability of modern computers. In the situation when the source strength varies with time (caused by accidental spill of pollutants or malfunctioning of the equipment etc.) the effect of dispersion should be included in the mathematical equation repre-senting the dynamics of BOD-DO balance in the river. The one dimensional mathematical model for the transport of BOD and its impact on DO, incorporating the advection and dispersion of entirely soluble BOD is considered as aB + u(x) aB = 1 (x) A(x) k,B at ax A(x) ax [ ax DC u(x) oc 1 a cat cox A(x) (x) A(x) flk1B + kr(Cg -C) ax x > 0, y > 0, t > 0 where B(x,t) and C(x,t) are concentration of BOD and DO respectively ; u(x), the mean cross sectional flow velocity; A(x), the cross sectional area; DL(x), the dispersion coefficient; k1, decay rate of BOD; kr, the coefficient of reaeration and C is the concentration of DO at saturation level. The model is applicable after the mixing length is over. The initial and boundary conditions associated with above partial differential equations are : B(x,0) = 0 and C(x,0) = Cs for x > 0 B(0,t) = 4(t) and C(0, t) = y (t) for x > 0 B(x, t) -+ 0 and C(x,t) --> CS as x-3 Two types of time dependent point sources at x = 0 are discussed. First case refers to an exponentially decaying source: (kW = Be exp (-kit) This occurs when BOD into the stream is coming from a stored waste water (undergoing exponential BOD decay) having no addition of waste water into the storage. In the second case a periodic source at the outfall is considered: 4)(t) = Bo 0 <t<T =0 t < t 2T where 2T is the time period, of the source. This situation represents the daily cyclic variation of domestic sewage and in the situation where an organic waste water, having uniform BOD, falling into the river varies in its flow rate with time in a cyclic manner. The analytical solutions generally require fewer numerical computations, they are applicable only in the idealized conditions of uniform velocity profiles uniform area of cross-section etc. The field problems, however require variable channel geometry and non-uniform velocity profile etc. Such practical problems can effectively be handled by numerical methods. To solve the above system of equation an explicit unconditionally stable scheme on a non uniform rectangular grid is presented. The grid points are computed • using the stream velocity profile. In this way the simplicity of fixed Eulerian grid is combined with the computational power of the Lagrangian approach. The scheme is free from numerical dispersion errors. It works within specified range of the time interval St computed apriori to ensure the positiveness of the coefficients in the scheme. However there is no restriction on spatial grid size. The existing DO sag models of Streeter-Phelps have become obsolete in the present day context of polluted streams in which a part of BOD removal necessarily takes place through sedimentation. This aspect is not accounted for in Streeter-Phelps model. The Bhargava's model incorporates linear removal of the settleable BOD and exponential decay of non-settleable BOD. A single expression in polynomialized form of 13hargava's model for critical DO deficit is iii developed. This expression is uniformly applicable irrespective of transition time. The polynomialized form of critical DO deficit has an additional advantage of evaluating the critical dissolved oxygen deficit directly without determining the time of occurrence of such deficit. A fairly good agreement with the theoretical values has also been observed. The dispersion models developed todate also do not account for settleable BOD. As such, these models are of little value in accurate prediction of BOD and DO in polluted streams. A mathematical model is, therefore, proposed to account for the settleable part of BOD. A source of BOD is assumed at the outfall in which a part of BOB is in settleable form and remaining BOD is in dissolved form. The waste is being discharged in to the river through an appropriately designed channel in which the sedimentation of settleable part is not allowed but the biochemical decay of total BOD is taking place., Thus, the total BOD at the out fall is decaying exponentially with time. This model is applicable when untreated or partially treated waste is coming into the river. The proposed mathematical model incorporates the advection dispersion and interaction of BOD-DO together with the sedimentation of the settleable part of BOD present in the waste. The proposed model is numerically solved with the help of finite difference scheme developed earlier. Due to non-availability of field data relating to this type of model, a theoretical but rational data is used. The numerical results are compared with the Bhargava's Kanpur summer data by taking uniform velocity and zero dispersion. The BOD distribution is same while DO distribution has minor differences due to truncation errors in finite differences. The distribution profiles for BOD and DO with time at various distances are drawn and compared with and without sedimentation effects. A conclusion is, therefore, drawn that if the partially treated or untreated waste is entering into the river in which the strength of BOD is varying with the time at the source, then sedimentation of settleable part must be taken into consideration along with dispersion to accurately predict the BOD-DO distribution in the river.