Mathematical modeling of BOD-DO in Rivers

By

Dr Babita Tyagi

Department of Mathematics

JRE Group of Institutions

Greater Noida(U.P)

Affiliated to UP Technical University Lucknow

Motivation

•Waste water is major cause of water pollution in natural waters

•Increased waste loading has been discharged directly to surface water without proper level of treatment.

•Mathematical modeling :

*Integrates physical , chemical and biological conditions for constructing complex environmental system

*provides problem-solving tools to predict the level of water quality in the receiving water

INTRODUCTION

The use of natural water present on the surface of the earth is manifold as it is required for the domestic, municipal,irrigation, industries, navigation and recreation etc. Unfortunately, the rapid growth of development in india paved the way for water pollution.

Water pollution has many sources. The most polluting of them are the city sewage and waste discharged into the river. The facilities to treat waste water are not adequate in any city in India. Presently only about 10% of the waste water generated is treated and the rest is discharged as it is into our water bodies. Due to this , pollutants enter groundwater , river and other water bodies. This results in the deterioratry condition of the water bodies which is reflected in many ways like fish mortality, foul smell etc.

Domestic sewage refers to waste water that is discharged from household. This sewage contains a wide variety of dissolved and suspended impurities. The disposal of domestic waste water is a significant technical problem. Sewage generated from the urban areas in India has multiplied manifold since 1947. The pollution by organic waste is the most common type of pollution and it’s effect is largely indirect. Organic wastes discharged into the rivers utilizes the DO of water through it’s aerobic decomposition. This depletes the oxygen of river water and makes it difficult for the biota to survive in oxygen-devoid water. The only aerobic life to survive in such water will be those organisms which are able to breath atmospheric air e.g. fish with lunges or having other accessory air-breathing organs or insects with similar adoption

In winters, when rivers are usually in full flow and cold enough to dissolve a large amount of oxygen, sewage can be accepted but in summer the river flows are low and at high temperature the discharge of oxygen robbing wastes can be fatal to aquatic life over large distances. The effect of low DO in surface water have been recognized for over a century and are reflected through unbalanced eco-system, through a wide variety of other water quality variables such as pH,TDS, salinity ect as given in the following table

Manifestation of problem Water use interferences

Water quality problem Parameter of high significance

1.Fish kills, Nuisance odor(H2S)”Nuisance” organism, radical change on Ecosystem.

Fishery, Recreation Ecological Health

Low DO ( Dissolved oxygen)

BOD,NH3,, Coliform,organic N2,organic solids, phytoplankton, DO, Heavy (or toxic trace) metals, physical parameters( turbidity etc)

2. Disease transmission ( Gastrointestinal) disturbance, eye irritation

Public water supply , Recreation

High Bacterial level Total colifom bacteria, Fecal colifom Bacteria, Fecal streptococci viruses.

3. Taste and odors, Blue-green algae , Aesthetic beach nuisance- algal mats “ pea soup” unbalanced ecosystem

Public water supply , Recreation,Ecological Health

Excessive Plant growth ( Eutrophication)

Nitrogen ,

Phosphorus,

Phytoplankton.

4.Carcinogens in Water Supply, Fishery closed unsafe toxic level ecosystem upset ( Mortality, reproductive impairment)

Public water supply, Fishery, Ecological System

High Toxic chemical levels Metals,

Radioactive substances

Pesticides,Herbicides, Toxic product chemicals

5. Plants health Agriculture High Concentration of TDS, Boron, high sodium ratio

TDS, Boron, Sodium Ratio

6. Various Manifestations such as taste and odors

Industrial High Bacterial level, high toxic chemical level

Depends on the kind of industry . For a inductry of soft drinks these parameters would be heavy metals and physical parameters( turbidity etc)

Table: *Principal Pollution Problems effected uses and associated water quality variables.

*The contents of table has been taken from references

DO of the river is the chemical constituent most frequently determined in observing the effect of oragnic pollution in streams. The BOD determination, which represents the amount of oxygen utilized in the aerobic stabilization of organic matter in a given period of time and at a specified temperature is used as an adjunct to the DO determination. Mathematical models are being used as an effective tool to predict DO conditions in river since past several years. These mathematical models are in the form differential equations representing various processes taking place in rivers.

Historical efforts mode in this field:

For the rivers not influenced by tidal action , the one dimensional model developed by Streeter and Phelp and later subsequent mathematical modeling by fair has become the standard tool for mathematical analysis of the BOD-DO problem.

The SP Model is a steady state model in which a point source of BOD is assumed at x=0. Advection is the only transport phenomena affecting the BOD and DO in rivers. The entire BOD is considered to be in soluble form which is decaying according to first order kinetics.

The coupled one dimensional differential equation representing BOD-DO Balance in rivers is given as follows.

This model was solved using the following initial condition.

(1.1)

(1.2)

(1.3)

In Eq. 1.1 and 1.2 the term represent the transport of BOD ( Denoted by B) and DO ( denoted by C) through the river velocity u . The steady uniform river velocity is averaged over cross-section. The term k1 B represents the first order decay of BOD. Since the demand is being fulfilled at the cost of river’s DO, the term is a sink for DO. The term kr(Cs-C) signifies the effect of reaeration. Reaeration is contributing to river’s DO, it would, therefore be a source for DO.

The one dimensional model presented by Streeter and Phelps is valid only after the mixing length is over.

The solution of Eq. 1.1, when substituted in Eq1.2 yields the uncoupled equation for DO as

(1.4)

(1.4)

The analytical solution of these equations is given by

Which is the famous Do sag equation formulated by Fair( ) Gunnesson ( ) and others.

(1.5)

The spoon-shaped curve in Fig. Known as the DO sag curve describes the self purification process in a stream. The engineers and Planners have interest in the lowest point in the curve.

At this point the uptake of oxygen by the BOD is just balanced by the input of oxygen from atmosphere. That is the process of de-oxygenation is balanced by reoxygenation. After this point the reaeration becomes dominant. The low point in the oxygen sag minimum DO attained in the stream and this minimum must satisfy the norms prescribed by the regulating legislative bodies for water pollution control.

It was later realized that this model is of little practical- importance as it does not include various other important phenomenon related to rivers, but this is considered as a landmark in the history of mathematical modeling of BOD-DO in rivers.

The assumption of SP model restrict the model capability to predict DO condition in stream where source strength varies with time and where the oxygen consuming waste contains settle able part also. To enable the effect of time varying source strength dispersion should also be included. Several researcher (Kousis,1983,Kousis et. al. 1983, Mcbribe and Rutherfords 1984) included the effect of dispersion also in their model.

Most of the dispersion model developed to date are uni-dimensional model. Though two and three dimensional models are presented by many authors in the situation when one – dimensional model is not applicable. But two and three dimensional modeling of transport requires a considerable amount of hydraulic data which has to be estimated. Rough estimates of parameters leads to the partial loss of accuracy gained.

More over various dispersion models were developed with the assumption that BOD is entirely in soluble form. These model do not account for BOD removal due to bioflocculation( followed by sedimentation) which normally takes place after partially treated or untreated sewage drains into the river.Tyagi et.al(1999) incorporated the effect of this BOD in their model along with the soluble BOD. The model is solved by finite difference method using an explicit scheme which is free from numerical dispersion.

In past several years, many authors have investigated the effect of dead zones on solute transport in stream and rivers(Hays et al (1996), Thackston and Krenkael1967, Thackston and Schnelle(1970)).

The travel time for solute carried through these zones may be significantly lesser than that of solutes traveling with the main body of water. Several approaches have been suggested to predict the impact of these zones on solute transport ( Rutherford(1994), Bencala and Walters(1983), Runkel(1998). Chapra(1999)) developed a steady-state model that explicitly considers the effect of transient storage on dissolved oxygen (DO) below a point source. This model is based on the following assumptions

(i) The stream’s hydrogeometry is constant.

(ii) Advection is the only transport mechanism

(iii) BOD is entirely in soluble form.

(iv) Plant –activity and SOD are negligible.

(v) Reaeration in storage zones is negligible.

(vi) Temperature in both zones ( i.e, storage zone and main channel)are same

The model was deliberately kept simple to understand the fundamental implication of transient storages.

Practically the rivers does not have constant channel geometry and dispersion may be an effective transport mechanism. In our work, a conceptual model is proposed to be developed which takes into account the following considerations

i) Dispersion effect are not negligible.

ii) Hydrogeometrical parameters are not constant.

iii) Velocity is varying in lateral direction.

The conceptual model is proposed to be solved using an effective finite difference scheme, which is free from numerical dispersion error and different type of boundary conditions involved.

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