Environmental Engineering Reference
In-Depth Information
body and in the sediment compartment. Thus, the oxygen saturation limit
c
DO,sat
depends on local circumstances and is not an universal (not even a terrestrial)
constant.
However, the reaeration process will make oxygen concentration converge to the
limit. Therefore in the most simple approach the difference from the limit
c
DO;sat
c
DO
determines the rate of DO-change aside from the kinetic parameter
k
2
.
The system (
9.1
) of two differential equations is quite simple to solve. The first
equation contains only one dependent variable (
c
BOD
) and has an exponential
function as analytical solution. With that solution one can regard the second
equation as a differential equation for
c
DO
.Together,both(
9.1
) form a linear
system. A method for the solution of general linear systems is presented below
(Chap. 18). The following lines are a special implementation for the Streeter-
Phelps system. The presented numerical approach can also be extended to non-
linear systems of equations, which may appear if there are complex parameter
dependencies.
T = 25; % maximum time [T]
k1 = 0.3; % deoxygenation rate coefficient [1/T]
k2 = 0.4; % reaeration rate coefficient [1/T]
DOsat = 11; % DO saturation [M/L^3]
BODin = 7.33; % initial BOD [M/L^3]
DOin = 8.5; % initial DO concentrations [M/L^3]
fBOD = 1; % natural BOD inflow [M/(L^3*T)]
N = 60; % discretization of time
%---------------------- execution ---------------------------------
% BOD = y(1), DO = y(2)
options = odeset('AbsTol',1e-20);
[t,y] = ode15s(@SP,[0 T],[BODin; DOin],options,k1,k2,DOsat,fBOD);
%---------------------- graphical output --------------------------
plot (t,y);
legend ('BOD','DO');
xlabel ('traveltime'); ylabel ('concentration');
grid;
%---------------------- function ----------------------------------
function
dydt = SP(t,y,k1,k2,DOsat,fBOD)
dydt = zeros(2,1);
dydt(1) = fBOD-k1*y(1);
dydt(2) = k2*(DOsat-y(2))-k1*y(1);
The complete code is included in the accompanying software under the name
'StreeterPhelps.m'
Input values for the example are adopted from Deaton and Winebrake (
2000
).
Deoxygenation and reaeration rate coefficients are given in 1/d. BOD inflow
