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Page 1: Mathematical modeling and parameter estimation for water quality management system.

Mathematical Modeling and Parameter Estimation for

Water Quality Management System.

Summer Internship report May 15, 2014– July 15, 2014

Student: Kamal Pradhan (12BTCSE04)

Program: B.Tech Computer Sc.

Department: SUIIT

Supervisor: Dr. Nihar Satapathy

Project name: AquaSense

SAMBALPUR UNIVERSITY INSTITUTE OF INFORMATION TECHNOLOGY

Submitted by: Kamal Pradhan, B.Tech CSE.

Guided By:

Dr. Nihar Satapathy H.O.D, Dept .of Mathematics,

Sambalpur University

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2 Acknowledgement

First I would like to thank Dr. Nihar Satapathy, H.O.D Dept. of Mathematics and P.I (AquaSense),

Sambalpur University for giving me the opportunity to do an internship within the organization. For me it was a

unique experience to be in Sambalpur and to study an interesting subject. It also helped to get back my interest in

ecological research and to have new plans for my future career. I also would like to thank all the people that

worked in the lab of ITRA in Sambalpur University Institute of Information Technology. With their patience and

openness they created an enjoyable working environment. Furthermore I want to thank all the Research fellows

and students, with whom I did the fieldwork. At last I would like to thank the ITRA Group, especially Dr. Nihar

Satapathy, Principal Investigator of the project AquaSense, to allow me to do this interesting internship.

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3 Abstract

This report describes various problem solving techniques in mathematical modeling for calculating

various parameters of water e.g. temperature, pH, Dissolved oxygen. A mathematical model provides

the ability to predict the contaminant concentration levels of a river. Here we are using an

advection-diffusion equation as our mathematical model. The numerical solution of equation

is calculated using Matlab & Mathematica. Parameter estimation is necessary in water

modeling to predict the different parameters of water at different point with minimal errors. So

here we use 2D & 3D interpolation technique for parameter estimation.

Introduction

This report is a short description of my two month internship carried out as a component of the B.Tech in computer science. The internship was carried out within the organization Sambalpur University Institute of Information Technology in from May 15-July 15 2014. Since my I am interested in Programming and quite acquainted with mathematical toolbox such as matlab and mathematica, the work was concentrated on solving complex mathematical problems programmatically. This internship report contains my activities that have contributed to project. In the following chapter a description of the organization ITRA and the activities is given. After this a reflection on my functioning, the unexpected circumstances and the learning goals achieved during the internship are described.

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4 Description of the internship

1. The organization ITRA

IT Research Academy (ITRA) is a National Programme initiated by Department of Electronics and Information Technology (DeitY), Ministry of Communications and Information Technology (MCIT), Government of India, aimed at building a national resource for advancing the quality and quantity of R&D in Information and Communications Technologies and Electronics (IT) and its applications at a steadily growing number of academic and research institutions, while strengthening academic culture of IT based problem solving and societal development. ITRA is currently operating as a Division of Media Lab Asia (MLAsia), a Section-25 not-for-profit organization of DeitY.

2. About the project AquaSense

To develop an indigenous, intelligent and adaptive decision support system for on-line remote

monitoring of the water flow and water quality across the wireless sensor zone to generate data

pertaining to utilization of water and raising alerts in terms of mails/messages/alarm following any

violation in the safety norms for the drinking water quality and usage of amount of water. This proposed

research objective is also to provide simple, efficient, cost effective and socially acceptable means to

detect and analyze water bodies and distribution regularly and automatically

to design and develop wireless sensing hardware for collecting hydraulic parameters like pressure, flow

and volume, and water quality parameters like Salinity, Color, pH, DO, Turbidity, Temperature,

Fluoride, Arsenic, Mercury, Lead, Selenium, Nitrate, Iron, Manganese and pathogens like Algal toxins

(cyan bacteria) etc

to design wireless sensor network zone architecture for drinking water flow and quality monitoring

to develop the interface modules (both hardware and software) for the wireless sensor nodes and probes

to develop an user interface for logging data after data fusion from the different sensor nodes

To design a database schema for storing on-line data received from the sensors.

a data collection and visualization infrastructure

to develop modeling and analysis tools (on-line estimation and prediction of the water distribution

system’s hydraulic state and leak/burst detection and localization)

to develop a Rule-base by incorporating the feedback of users to include the quality perception of users

based on the locality and preferences through machine learning algorithms

to develop a knowledge-base for different regions and applications

to develop an expert system for adaptive setting up of new bench mark for water quality of drinking

water and other usage

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5 3. Mathematical Modeling and finding numerical solutions

Mathematical Model for the Concentration of Pollution using Advection-

Diffusion equation

A mathematical model provides the ability to predict the contaminant concentration levels of a

river. We present a simple mathematical model for river pollution. The model consists of a

pair of coupled reaction diffusion-advection equations for the pollutant and dissolved oxygen

concentrations, respectively. We consider the steady state case in one spatial dimension. For

simplified cases the model is solved analytically by considering the case of zero dispersion,

that’s mean ( Dp=0 and Dx=0).

The standard advection-diffusion-equation may be written as follows:

C: Concentration of pollutant

D: Diffusion Coefficient

u: Mean flow velocity

x: Position

t: Time

Solving the equation through programmatically:

Mathematica Code

Here we are solving the advection diffusion equation where the time is varying from 0-2

seconds and the position is varying from –pi to pi.

sol = NDSolve[{𝐷[𝑐[𝑡, 𝑥], 𝑡] == 0.5𝐷[𝑐[𝑡, 𝑥], 𝑥, 𝑥] + 𝑐[𝑡, 𝑥]𝐷[𝑐[𝑡, 𝑥], 𝑥], 𝑐[𝑡,−Pi] == 𝑐[𝑡, Pi] ==0, 𝑐[0, 𝑥] == Sin[𝑥]}, 𝑐, {𝑡, 0,2}, {𝑥,−Pi, Pi}];

By plotting solutions evaluated from the above equation we obtain the following graphs.

Plot3D[Evaluate[𝑢[𝑡, 𝑥]/. First[sol]], {𝑡, 0,2}, {𝑥,−Pi, Pi}, PlotRange → All]

Fig. 1. Graphs of advection diffusion equation

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Now if we vary the position and time over a period of t=30 with density =0.7 and

concentration of pollutant.

Solution of advection dispersion equation using matlab

Fig. 2 Graph of advection diffusion equation with varying parameters

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8 Matlab functions to solve the equations

1.

%Boundary Condition

function [ p1,q1,pr,qr ] = pdebc( x1,cl,xr,cr,t )

p1=cl-1; q1=0; pr=cr; qr=0;

end

%Boundary Condition

function [ p1,q1,pr,qr ] = pdebc( x1,cl,xr,cr,t )

p1=cl-1; q1=0; pr=cr; qr=0;

end

%Initial Condition

function [ c0 ] = pdeic( x )

c0=0;

end

% Main Function

function [ g,f,s ] = pdefun( x,t,c,DcDx ) %PDEFUN Summary of this function goes here % Detailed explanation goes here

D=2; g=4; f=D*DcDx; s=0;

end

Fig. 3 Graph obtained by varying x= (0, 2.5, 200) & t= (0, 5,100)

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4. Parameter Estimation in Modeling

Parameter estimation plays a critical role in accurately describing system behavior through

mathematical models such as statistical probability distribution functions, parametric dynamic models,

and data-based models.

In the mathematical field of numerical analysis, interpolation is a method of constructing new data

points within the range of a discrete set of known data points.

As water body is dynamic i.e. it contains more than coordinate so we cannot use linear interpolation to

estimate a particular parameter in a given point.

Let us assume that we place sensors in the upper surface of water so now the sensors are in a 2

dimensional coordinate system (Fig. 4).

Here we can see that the river is a regular body i.e. square in shape. The sensors are placed in

the body such that it exactly covers the river. Here we cannot use linear interpolation as a single

point in the grid is surrounded by 4 sensors. Therefore the value at a single point depends upon

the value that is accused by the neighboring sensors. The solution to this problem is bilinear

interpolation or gridded interpolation.

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Bilinear interpolation is used when we need to know values at random position on a regular 2D

grid. The key idea is to perform linear interpolation first in one direction, and then again in the

other direction. Although each step is linear in the sampled values and in the position, the

interpolation as a whole is not linear but rather quadratic in the sample location.

Here the input data of temperature is between points -5 to 3 with 0.25 as interval but we get the

interpolated data with a 0.125 interval. so by bilinear interpolation we can easily estimate a

parameter in particular point.

Fig. 5 Graph of 2d interpolation for 2d water body for temperature.

Input Data

Interpolated

data

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If we consider river as a 3D model i.e. it has x, y and z coordinate. We cannot use a bilinear

interpolation for parameter estimation. Here we will use 3D interpolation for paramet

estimation.

3D interpolation Graphs

Input data Output data

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The following data were collected by T. N. Tiwari and S. N. Nanda in the year 1999.