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Now-a-days environmental degradation has increased throughout the world due to the inclusion of various pollutants into the water bodies. The issues of pollutant transport in the water-body either through continuous sources of pollution or instantaneous sources have a significant impact on environment in the wake of economic and industrial development of the countries. The numerical models are efficient tools to meet the challenges of accurate predictions of the distribution of contaminant concentrations in the rivers.
The objective of the study is 1) to develop 1D numerical model and 2) to simulate the concentration profile at downstream sections of 5km, 20km of a straight rectangular channel of defined geometry (B=10m, H=0.4, n=0.01, S0=0.0005, M0= 10kg).
In this connection, finite difference methods such as Lax-Wendroff scheme method has been applied for solving one dimensional pure advection equation having the initial and boundary conditions and one dimensional advection-dispersion equation has also been considered to solve by Lax-Wendroff scheme.
The two different schemes have been simulated and tested through MATLAB codes by using constant velocity but changing time step and space step for varying courant number by taking three test cases for specific time period and domain.
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In this report, the accuracy and stability of numerical solutions produced by two different schemes has been examined. An analytical solution by Microsoft Excel for the concentration profile of the pollutants at different locations of the given channel geometry has been produced using 1D advection dispersion equation. Finally a comparative statement of the concentration profile of the pollutants is made for numerical solutions of different methods and analytical solutions using tables and graphs.
It has been a great experience for me to study, work and submit a research paper entitled “Develop 1D Advection Dispersion numerical model and simulate the concentration profile at downstream sections of 5km, 20km of a straight rectangular channel of defined geometry (B=10m, H=0.4, n=0.01, S0=0.0005, M0= 10kg)” for fulfillment of the course Mathematical Model of Water Environment.
I would like to convey my gratitude and sincere thanks to PROFESSOR YIQINGGUAN for his great effort of giving knowledge on analytical and numerical methods for solutions or prediction of contaminant mass transport and enable me to insight the different mixing mechanisms in reservoirs, rivers and estuaries .
It is actually quite hard to know how the pollution displaces itself in water because it depends of a lot of parameters (for example, weather, wind, turbidity, flow...).
The main difficulty is the propagation of pollution in water in 3D dimension. The actual document will show and 1D dispersion with advection equation. To do this, we will use the software Mat lab. For this exercise, we will find the solution using the Lax-Wenderoff method (1D) and give an analysis of the results.
For this exercise, we have to use this formula:
Finite difference method is employed to solve ADE
1D advection dispersion equations
Finite difference method is employed to solve ADE
By using LW scheme, it is given below:
Moreover, we have those conditions:
1D Advection-Dispersion problems:
One straight rectangular channel is introduced 10 kg tracer at one section fully mixed. H=0.4m, B=10m, S0=0.0005. n=0.01. Simulating the time variation of concentration at downstream section of 5 km and 20 km, respectively by using numerical methods
Mo=10kg
B = 10 m (Width)
H = 0.4 m (Height)
S0 = 0.0005 (Slope)
n =0.01
L = 25km (Length)
To simulate the concentration, we will use finite difference schemes at sections 5 km and 20 km
The Lax–Wendroff method, named after Peter Lax and Burton Wendroff, is a numerical method for the solution of hyperbolic partial differential equations, based on finite differences. It is second-order accurate in both space and time. This method is an example of explicit time integration where the function that defines governing equation is evaluated at the current time
Suppose one has an equation of the following form:
Where x and t are independent variables, and the initial state, u(x, 0) is given.
We use the Mat lab software to insert a code which should give us a result with a graphic.
On it, we would be able to understand how the pollutant displaces itself in a 1D direction in the water.
Matematical Model
Solution for 1D Advection-dispersion Equation with Lax-Wendroff Scheme
Initial Data
B = 10; %Channel Width [m]
H = 0.4; %Channel Height [m]
So = 0.0005; %Channel Slope [-]
n = 0.01; %Manning Coefficient [-]
dx = 1000; %Delta x [m]
dt = 100; %Delta s [s]
n_time = 900; %Number of Time Steps [-]
i_space = 200; %Number of Space Steps [-]
To=1000*9.81*H*So; %Bed Shear Stress [N/m^2]
Ustar=sqrt(To/1000); %Friction Velocity [m/s]
Kx=5.93*H*Ustar; %Dispersion Coefficient[m^2/s]
U = (1/n)*(H^(2/3))*So^(0.5)*1; %Mean Velocity [m/s]
alpha = U*dt/dx; %Courant Number
betta=Kx*dt/(dx^2); %Betta
%% Initial Condition
for i = 1:i_space
c(i) = 0; end for i = 5:15;
c(i) = 10; end for i = 40:60;
c(i) = 10*sin(pi*i/(20)).^2; end for i = 1:i_space
C(i,1) = c(i); end Numerical Solution (Lax-Wendroff Scheme)
for n = 2: n_time
for i = 2: i_space-1;
C(i,n) = C(i,n-1)-alpha/2*(C(i+1,n-1)-C(i-1,n-1))+(alpha^2)/2*(C(i+1,n-1)-2*C(i,n-1)+C(i-1,n-1))+betta*(C(i+1,n-1)-2*C(i,n-1)+C(i-1,n-1));
C(1,n) = 0;
C(i_space,n) = C(i_space-1,n);
Plotting
s_node = 1:1:i_space; Space node
t_node = 1:1:n_time; Time node
x = s_node*dx/1000; X (km)
t = t_node*dt/3600; t (hour)
We can choose to plot in (x or s_node) and in (t or t_node)
Initial = c(1:i_space); %Initial Conditon plot (x,Initial,'r'); %Plotting Initial Condition in x (m)
title('X vs Concentration');
xlabel('x (km)'), ylabel('Concentration (mg/L)');
axis auto hold on
Sol_t = C(:,500); %Solution at any x at time step 500
plot(x,Sol_t,'g'); %Plotting at any x at time step 500
hold off
X_5= C(20,:); %Solution at 5 km at any time step
plot(t,X_5,'b-'); %Plotting at 5 km at any time step
title('Concentration at 5 km')
xlabel('t (hour)'), ylabel('Concentration (mg/L)');
axis auto
X_20 = C(40,:); %Solution at 20 km at any time step
plot(t,X_20,'b-'); %Plotting at 20 km at any time step
title('Concentration at 20 km')
xlabel('t (hour)'), ylabel('Concentration (mg/L)');
axis auto
print('water model','-dpng');
Comparison of concentration between the numerical solution and analytical solution
From the above graph read colour show Analytical solution
Green show model solution
Concentration at 20km
Concentration at 5km
Unstable calculation by excel of concentration at 5km and 20km
We can see that the concentration at 5 kilometres is higher than at 20 kilometres and appears quicker. So the more far we are, the less the pollutant is concentrated.
The objectives for many countries is to get back a good water quality and for others to Stop the deterioration of the rivers. To understand more about what the pollution becomes after arriving in the water. For this, we use the advection-equation to see how the pollution displaces itself. For this exercise, we use on the Matlab software the Lax-Wenderoff scheme for the advection-dispersion solution.
As results, we saw that the concentration is less high the further we go. Moreover, the pollutant stays longer due to the dispersion effect. So we can conclude that the pollutant sees its concentration decrease in space and time.
With that kind of researches, we can understand how the dispersion works and use that to solve pollution problems, construction of dams and dinks, in objectives of efficiency, robustness and prevention.
Numerical Modeling of Pollutant Transport in a Rectangular Channel Using Finite Difference Methods. (2024, Feb 22). Retrieved from https://studymoose.com/document/numerical-modeling-of-pollutant-transport-in-a-rectangular-channel-using-finite-difference-methods
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