Solution to Problem 1.3

Question 1 

 

t (day)

t (sec)

Cmax

0.6 Cmax

(cm)

1

86400

13.5

8.1

3.0

4

345600

6.8

4.1

6.0

 

 

 

 

 

Assuming you released the dye in a very thin horizontal layer, such that initially  = 0, then:

 

Thus, our two estimates of D are:

 

D1 = (3.0 cm)2 / 2(86400 s) = 5.2 x 10-5 cm2s-1.

and

D2 = (6.0 cm)2 / 2(345600 s) = 5.2 x 10-5 cm2s-1.

 

Therefore, the estimated diffusion coefficient of the dye is 5.2 x 10-5 cm2s-1.

 

 

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Question 2

 

        We must consider the boundaries of the tank when the cloud width (4) is 40 cm.

 

That is, when:

4(2Dt)1/2 = 40 cm

(i.e. when Dt = 50 cm2)

 

Using our value of D above, the boundaries of the tank become important when

t = 50 cm2/ 5.2 x 10-5 cm2s-1 = 9.6 x 105 s = 11.1 days.


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