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Lecture 7: Pollutant Transfer in a Stream
Introduction.
Consider a river that has been polluted upstream.
The
concentration (amount per volume) will decay and disperse downstream. We would like
to predict at any point in time and in space the concentration of the pollutant. The model
of the concentration will also have the form newy = A oldy + b.
Pollution in streams, lakes and underground aquifers has become a very serious
common concern. It is important to be able to understand the consequences of possible
pollution and to be able to make accurate predictions concerning "spills" and future
"environmental" policy.
Perhaps, the simplest model for chemical pollutant is based on chemical decay,
and one model is similar to radioactive decay. A continuous model is ut = - d u where d
is a chemical decay rate. A discrete version is uk+1 = uk + ∆t (-d) uk, and for stability this
requires 1- ∆t d > 0.
Here we will introduce a second model where the pollutant changes location
because it is in a stream. We assume the concentration will depend on both space and
time. The space variable will only be in one direction, which corresponds to the direction
of flow in the stream. If the pollutant were in a deep lake, then space should have all
three directions.
Model.
Discretize both space and time, and let the concentration equal to u(i∆x,
k∆t) be approximated by uik where ∆t = T/maxk, ∆x = L/n and L is the length of the
stream. The model will have the general form
change in amount ≈ (amount entering from upstream)
- (amount leaving to downstream)
- (amount decaying in a time interval).
This is depicted in the figure below where the time step has not been indicated
entering volume = A vel ∆t
A
vel > 0
x - ∆x
Figure:
x
Polluted Stream
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Assume the stream is moving from left to right so that the velocity is positive, vel
> 0. Let A be the cross sectional area of the stream. The amount entering the left side of
the volume A ∆x (vel > 0) is
A ∆t vel ui-1k.
The amount leaving the right side of the volume A ∆x (vel > 0) is
-A ∆t vel uik.
Therefore, the change in the amount from the stream's velocity is
A ∆t vel ui-1k - A ∆t vel uik.
The amount pollutant in the volume A ∆x at time k∆t is
A ∆x uik.
The amount of the pollutant that has decayed, dec is decay rate, is
-A ∆x ∆t dec uik.
The change during the time interval in the amount of pollutant in the small volume A ∆x:
A ∆x uik+1 - A ∆x uik = A ∆t vel ui-1k - A ∆t vel uik - A ∆x ∆t dec uik.
Now, divide by A ∆x and explicitly solve for uik+1.
Explicit Finite Difference Model of Flow and Decay of a Pollutant.
uik+1 = vel (∆t/∆x) ui-1k + (1 - vel (∆t/∆x) - ∆t dec) uik where
(1)
i = 1,...,n-1,
k = 0,...,maxk-1,
ui0 = given for i = 1,...,n-1 and
(2)
k
u0 = given for k = 1,...,maxk.
(3)
Equation (2) is the initial concentration, and (3) is the concentration far upstream.
Equation (1) may be put into the matrix version of the first order finite difference
method. For example, if the stream is divided into four equal parts, then n = 4 and (1)
may be written as either three scalar equations, or one 3D vector equation:
u1k+1 = vel (∆t/∆x) u0k + (1 - vel (∆t/∆x) - ∆t dec) u1k
u2k+1 = vel (∆t/∆x) u1k + (1 - vel (∆t/∆x) - ∆t dec) u2k
u3k+1 = vel (∆t/∆x) u2k + (1 - vel (∆t/∆x) - ∆t dec) u3k
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uk+1 = A uk + b where d = vel (∆t / ∆x) and c = 1 − d − dec ∆t
u1k +1   c
 k +1  
 u2  =  d
u3k +1  


c
d
k
k
 u1   du0 

 k 
  u2  +  0  .
c  u3k   0 
(4)
An extremely important restriction on the time step ∆t is required to make sure the
algorithm is stable. For example, consider the case n = 2 where the above is a scalar
equation, and we have the simplest first order finite difference model. Here a = 1 - vel
(∆t/∆x) - dec ∆t. If a = 1 - vel (∆t/∆x) - dec ∆t > 0 and vel, dec > 0, then this simple
condition will imply that the matrix products Ak converge to the zero matrix. This will
make sure there is no blowup, provided the source terms are bounded.
Stability Condition for (1).
1 - vel (∆t/∆x) - dec ∆t and vel, dec > 0.
In the case where dec = 0, then a = 1 - vel (∆t/∆x) > 0 means the entering fluid
must must not travel more than one space step. This is often called the Courant
condition on the time step.
Method.
In order to compute all uik+1, which we will henceforth denote by u(i,k+1),
we must use a nested loop where the i loop (space) is inside and the k loop (time) is the
outer loop. In this flow model u(i,k+1) depends directly on the two previously computed
u(i-1,k) and u(i,k). This is different from the heat diffusion model which requires an
additional value u(i+1,k) and a boundary condition to the right side.
Implementation. We use a Matlab code for the explicit flow and decay model of a
polluted stream. The first graph is the concentration versus time and space. Here the
initial concentration was a trig function upstream and zero downstream. The farthest
upstream location always had zero concentration. One expects the location of the
maximum concentration to move downstream and to decay.
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Matlab Code (flow1d.m)
% Flow in a Stream
clear;
% Length of the Stream
L = 1.0;
% Duration of Time
T = 20.;
K = 200;
dt = T/K;
n = 10.;
dx = L/n;
vel = .1;
decay = .1;
% Initial Concentration
for i = 1:n+1
x(i) =(i-1)*dx;
u(i,1)
=(i<=(n/2+1))*sin(pi*x(i)*2)+(i>(n/2+1))*0;
end
% Upstream Concentration
for k=1:K+1
time(k) = (k-1)*dt;
u(1,k) = -sin(pi*vel*0)+.2;
end
% Time Loop
for k=1:K
% Space Loop
for i=2:n+1;
u(i,k+1) =(1 - vel*dt/dx -decay*dt)*u(i,k) +
vel*dt/dx*u(i-1,k);
end
end
mesh(x,time,u')
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Figure:
Concentration as Function of (t,x)
The second graph is a contour graph of the concentration in the time versus space
plane. Here one can see time-space regions that have concentration of the pollutant
between certain levels.
contour(x,time,u')
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18
16
14
12
10
8
6
4
2
0
0
0.1
Figure:
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Contours of Concentration
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The following Matlab m-file will produces a frame by frame "movie" which does
not require a great deal of memory. This code generates graphs of the concentration
versus space for a sequence of times. In the pollution model it shows the pollutant
moving down stream and decaying.
Matlab Code (mov1d.m)
flow1d;
lim =[0 11 0 1];
for k=1:5:150
plot(u(:,k))
axis(lim);
k = waitforbuttonpress;
end
Another way to view the output is collection of concentration versus time plot
where one can observe the peak concentration moving downstream.
plot(x,u(:,1),x,u(:,51),x,u(:,101),x,u(:,151))
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0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
Figure:
0.2
0.4
0.6
0.8
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Concentrations for Increasing Time
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In the following calculation we let vel = 1.3, and this, with the same other
constants, violates the stability condition. For the time step of 1/10 and the space step of
1/10, a flow rate equal to 1.3. means that the pollutant will travel 1.3/10 units in space
which is more than one space step.
flow
Figure:
Assessment.
Unstable Computation with vel ∆t > ∆x
The discrete model is accurate for suitably small step sizes.
k+1
A ∆x ui - A ∆x uik = A ∆t vel ui-1k - A ∆t vel uik - A ∆x ∆t dec uik.
Divide by A∆x ∆t
(uik+1 - uik)/∆t = vel (ui-1k - uik)/ ∆x - dec uik.
Approximate the first order partial derivatives
ut(i∆x, k∆t + ∆t/2) = -vel ux(i∆x - ∆x/2, k∆t) - dec u(i∆x, k∆t).
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The dispersion of the pollutant is a continuous process, which could be modeled by a
partial differential equation and initial and boundary conditions:
ut = -vel ux - dec u,
u(x, 0) = given and
u(0, t) = given.
This is analogous to the discrete model in (1), (2) and (3). Like the heat models the step
sizes should be carefully chosen.
Often it is difficult to determine the exact values of the constants vel and dec.
Exactly what is the effect of having measurement errors, say of 10%, on constants vel,
dec or the initial and boundary conditions? What is interaction of the measurement errors
with the numerical errors? The flow rate, vel, certainly is not always constant.
Moreover, there may be fluid flow in more than one direction.
Homework.
1.
2.
3.
4.
Write a computer program and observe the consequences of not satisfying
the stability condition.
Experiment with the mesh size and observe convergence as the mesh size
decreases.
Vary the flow rate, vel, and the decay rate, dec. Explain your computed
results.
Consider the 3x3 A matrix in (4). Observe Ak for different values of vel
so that the stability condition either does or does not hold.