6.18

A mass balance for a pollutant in a well-mixed lake can be written as

V * dc/dt = W - Q*c - k * V * sqrt(c)

Given the parameter values V = 1 * 10^6 m^3 , Q = 1 * 10^5 * m^3/yr,W = 1 * 10^6 g/yr, and k = .25 m^0.5/g^0.5/yr, use the modified secant method to solve for the steady-state concentration. Employ an initial guess of c = 4 g/m^3 and delta = .5. Perform three iterations and determine the percent relative error after the third iteration.

Matlab Code:

V = 1.0 * 10^6; % m^3 

Q = 1.0 * 10^5 ; % m^3/yr,

W = 1.0 * 10^6 ; %g/yr

k = .25; % m^0.5/g^0.5/yr,

c = -20:20;

f = @(c) W - Q .* c - k * V * sqrt(c);

hold on

plot(c, f(c))

c = 4; % g/m^3

delta = .05; 

for ii = 1:5

    plot(c, f(c), '*r')

    

    c_new = c - (delta * c * f(c))/(f(c + delta * c) - f(c));

    c = c_new;

end

hold off

percent_relative_error = ((c_new - c)/c_new) *100

The resulting percent relative error was 0