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.

1. 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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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.

3. AquaSense, Internship 2014
3Abstract
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.

4. AquaSense, Internship 2014
4Description 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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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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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)

9. AquaSense, Internship 2014
94. 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.

10. AquaSense, Internship 2014
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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.