After solving parts a and b assign the following parameter

Question

The discharge of a large concentration of organic matter into an otherwise healthy stream produces a phenomenon known as the oxygen sag. The introduction of a large amount of fuel in the form of the organic matter causes a sharp increase in the productivity of the indigenous microorganisms. Since microbes utilize dissolved oxygen in the process, the net result is a decrease in the oxygen concentration, A, as a given "slug" of water (some fixed volume) moves downstream. Assuming no other sources of organic matter exist downstream, eventually the oxygen concentration of the water rebounds towards its saturation level, A_sat. However, before this occurs, the oxygen concentration reaches a minimum ("sag") value at some time after the discharge of organic matter into the moving slug of water.

The oxygen sag phenomenon is modeled mathematically by the following differential equations.

dB/dt = -k₁B                        (Eq A)

where B  = the concentration of organic matter in mg/L

k₁ = the rate constant of organic matter concentration decay

k (A_sat -  A) - k₁ B = dA/dt                       (Eq B)

where A_sat = the saturation concentration of oxygen in water (a constant at a given water temperature)

A = the dissolved oxygen concentration in water, a function of time

k = the constant mass transfer coefficient of water between the atmosphere and water

h = the mean depth of the stream

The mass transfer coefficient, k, can be estimated by the following: k = (Dv/h)^1/2

where D the molecular diffusivity of oxygen in water (a constant at a given water temperature)

v = the constant average linear water flow velocity

A. Find the general solution to Eq A for B as a function of time. Then find the particular solution for the initial condition B(0) = B₀.

B. Substitute the particular solution for B into Eq B and find the general solution to Eq B for the oxygen concentration, A as a function of time. Find the particular solution for the initial condition A(0) = A_o.

C. After solving parts A and B, assign the following parameter values and then use differential calculus to determine how long it will take for the oxygen concentration to reach the "sag" value. (Hint: Think about the slope of the tangent line). Then graph the concentration of oxygen as a function of time over the interval 0 days < t < 10 days.

A₀ = 6.17 mg/L                D = 1.8 le cm2/sec
B₀ = 25 mg/L                   v = .12 m/sec
k_{1 0.3 / day}                      h = .76m
A_sat = 8.263 mg/L