The first order differential equation below describes the washout of a non-reacting material ("component A"), from a well-mixed stirred tank, lake, or room.

dCA/dt = 1/tau[CAin - CA]

where CA is the concentration of A in the tank and the effluent stream (mg/mL).

where CAin is the concentration of A in the inlet stream (mg/mL).

tau is the residence time of the tank (= volume V/flow, q).

V is the tank volume (L)

q is the influent and effluent outlet flow rate (L/sec).

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To integrate the equation, we assume an initial concentration of A (CA0) is 10 mg/mL, an influent A concentration of 0, a tank volume of 10,000,000 L, and a flow rate of 1,000 L/sec.

(notice that this system also describes the air quality in a house or auditorium, or city airshed, purification of a poisoned reservoir, or a chemical reactor).

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The analytical solution is:

CA = CAin + [CA0 - CAin] e^-t/tau

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a. Are the units of this equation OK?

b. Do the long and short-time limits make sense?

c. Use the Euler method to solve the equation with a few different time steps.

d. At what time has the initial concentration fallen by half? Does this make sense?

e. Plot the numerical results against the analytical solution given above.

f. Make a plot of (Canalytical-CEuler)/Canalytical vs. time, out to a reasonable amount of time (enough to show the dynamics well).

g. Redo the calculation of part c and plot of part f with a step size of 2 seconds.

dCA/dt = 1/tau[CAin - CA]

where CA is the concentration of A in the tank and the effluent stream (mg/mL).

where CAin is the concentration of A in the inlet stream (mg/mL).

tau is the residence time of the tank (= volume V/flow, q).

V is the tank volume (L)

q is the influent and effluent outlet flow rate (L/sec).

---------------------

To integrate the equation, we assume an initial concentration of A (CA0) is 10 mg/mL, an influent A concentration of 0, a tank volume of 10,000,000 L, and a flow rate of 1,000 L/sec.

(notice that this system also describes the air quality in a house or auditorium, or city airshed, purification of a poisoned reservoir, or a chemical reactor).

---------------------

The analytical solution is:

CA = CAin + [CA0 - CAin] e^-t/tau

---------------------

a. Are the units of this equation OK?

b. Do the long and short-time limits make sense?

c. Use the Euler method to solve the equation with a few different time steps.

d. At what time has the initial concentration fallen by half? Does this make sense?

e. Plot the numerical results against the analytical solution given above.

f. Make a plot of (Canalytical-CEuler)/Canalytical vs. time, out to a reasonable amount of time (enough to show the dynamics well).

g. Redo the calculation of part c and plot of part f with a step size of 2 seconds.

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