Assignment Title: Accidental sewage discharge into a river Case Solution And Analysis, HBR Case Study Solution & Analysis of Harvard Case Studies

INTRODUCTION

The problem associates with our daily life that harmful polluted wastes are discharged in the river and lakes. Polluted material is discharge in the river or lakesinnature, transport take place in fluids through the combination of advection and diffusion.

The waste is diffused in it which causes danger for humans as well as animals. In order to find the degree of harmfulness of pollution, it is important to find concentration rate and diffusion rate of pollution by numerical method. By numerical method,the diffusion and derived solutions are used to predict diffusive transport in stagnant ambient conditions. We use crank Nicolson method to find and approximate solution of system(Tang, ‎1993).

NUMERICAL METHODS

We have to implementation of such methods which can easily compute the value of rate of concentration of pollutant. The governing equation is

Here “c” be the concentration of pollutant.

“U” be the velocity of river.

“D” be the diffusion coefficient.

Discretization:

The equation needs to be discretizedto convert it to a set of points. Discretization will be done for finite differences, which should be located between mesh points(Davis, ‎1984).

For solving the problem there aretwo schemes for finding the rate of concentration with respect to distance. Thefirst scheme is called explicit difference and the second is implicit difference.
We implement implicit difference scheme in our matlab script.

This method is more convenient to solve such type of system

MATHEMATICAL EQUATIONS FOR IMPLICIT SCHEME:

We implies implicit scheme for solving the problem.

C_iⁿ⁺¹=C_iⁿ – Udt (C_iⁿ⁺¹- C_i^n-1)/2dx +D(C_iⁿ⁺¹-2Ciⁿ + Cin^-1)/dx²

We take the parameter form data

L=50m

D=1

U=1

Dt =1

Dx=L/m

Here m be the number of mesh point.

Now we have

C_iⁿ= 0C_iⁿ⁺¹ -0.6604C_i-1ⁿ⁺¹ -0.0601 C_i+1ⁿ⁺¹ +1.7205 C_iⁿ⁺¹

The above equation represents the coefficient of concentration vector in the time t=n+1 however,the mesh point is increasing;this known asthe central difference equation. The method implies the backward difference scheme to find out the previous point.

BOUNDARY CONDITION FOR IMPLICIT SCHEME:

Given that at the western boundary, the flow concentration Ci at these points is 0 at all times.

C_i=0
Also, we have given that at the eastern outflow boundary, the boundary condition is complex. As a close approximation, we will assume the second spatial derivative of the concentration is zero to the boundary

dC/dx² = 0

Advantages and Disadvantages of Finite Difference Explicit and implicit:

If we consider initial-boundary value problems for parabolic functional differential equations, then the explicit difference schemes of Euler type and implicit difference methods are investigated.
There is comparison of explicit and implicit difference schemes for parabolic functional differential equations.
If sufficient conditions for the convergence of approximate solutions are given and comparisons of the methods are present then the explicit scheme is appropriate in this case. While the condition of convergent is not present therefore,the best scheme for this problem is implicit scheme.

Assignment Title Accidental sewage discharge into a river Case Solution

For Explicit scheme we can say that direct computation of the dependent variables can be made in terms of known quantities.

Advantage of finite difference method is easy to solve simple differential equations because it is easy to implement.

If we come towards the disadvantage of finite difference, the method is difficult to implement on complex and complicated differential Equations.

Another disadvantage is that Explicit Model requires condition for stability as in this assignment for stability it required(Gaidamauskaite, 2007).

Uδt 2D ≤ 1 (18)

Dδt δx2 ≤ 1 2

On the other hand,Implicit Model does not require any condition for stability as it is numerically stable and convergent, which has a bid advantage and beauty of this method

Description

The 3d model of graph represents the increment in concentration over time and distance. From the time 1.5 to 2, the concentration goes down after its slope of concentration increases more rapidly in time from 3.5 to 4.

This show that the diffusion process is slow initially and it increases with time, whereas the upper portion of graph shows that concentration increases but diffusion also increases. As the distance increases the slope of concentration is almost linear because the ratio of concentration and diffusion is constant almost throughout the time.

Solution Method:

Description of script:

In the Matlab coding, we took the coefficient from A matrix and corresponding matrix B and solved through Crank Nicolson method.We described each code line in the programming as we used different command and mathematical operator to solve equation and used nested loop to print and load 2 dimensional values.

We initialized all constants and made a vector for the prediction of response of the advection diffusion equation. In the next step, we setup the vectors for calculating the next value of in another time

In the last part of the script is graphical representation of concentration verses distance and concentration verses time. The graphical representation shows the concentration with respect to time and distance; it increases with respect to time and distance. Moreover, 2D graph displays concentration with respect to time.................

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