Water flows between the North American Great Lakes as depicted in Fig. P13.12. Based on mass balances, the following differential equations can be written for the concentrations in each of the lakes for a pollutant that decays with first-order kinetics:

where k = the first-order decay rate (/yr), which is equal to 0.69315/(half-life). The constants in each of the equations account for the flow between the lakes. Due to the testing of nuclear weapons in the atmosphere, the concentrations of strontium-90 (⁹⁰Sr) in the five lakes in 1963 were approximately {c} = {17.7 30.5 43.9 136.3 30.1}T in units of Bq/m³. Assuming that no additional ⁹⁰Sr entered the system thereafter, use MATLAB and the approach outlined in Prob. 13.11 to compute the concentrations in each of the lakes from 1963 through 2010. Note that ⁹⁰Sr has a half-life of 28.8 years.

Data From Problem 13.11

A system of two homogeneous linear ordinary differential equations with constant coefficients can be written as

If you have taken a course in differential equations, you know that the solutions for such equations have the form y_i = ce^λt where c and  are constants to be determined. Substituting this solution and its derivative into the original equations converts the system into an eigenvalue problem. The resulting eigenvalues and eigenvectors can then be used to derive the general solution to the differential equations. For example, for the two-equation case, the general solution can be written in terms of vectors as

where {ν_i} = the eigenvector corresponding to the i^th eigenvalue (λ_i) and the c’s are unknown coefficients that can be
determined with the initial conditions.
(a) Convert the system into an eigenvalue problem.
(b) Use MATLAB to solve for the eigenvalues and eigenvectors.
(c) Employ the results of (b) and the initial conditions to determine the general solution.
(d) Develop a MATLAB plot of the solution for t = 0 to 1.

Transcribed Image Text:

(2- Ax² p²) -1 0 0 -1 (2 - Ax² p²) -1 0 0 -1 (2- Ax² p²) -1 yı Y2 (2-4x²0²) 33². 0 −1 Y4 = 0 =