Problem 41. The cornea of the eye must be transparent, so it can contain no bloo

Question

Problem 41. The cornea of the eye must be transparent, so it can contain no blood vessels. (Blood absorbs light.) Oxygen needed by the cornea must diffuse from the sur- face into the corneal tissue. Model the cornea as a plane sheet of thickness L = 500pm. The oxygen concentration, C, is governed by a one-dimensional steady-state diffusion equation dec dx? = Q. Assume the cor nea is consuming oxygen at a rate Q 4 x 1022 molecule m-3 s-1 and has a diffusion constant = 3 x 10-9 m2 s-I ·The rear surface of the cornea is in contact with the aqueous humor, which has a uniform oxygen

Explanation / Answer

d2C/dx2 = Q/D

=> dC/dx = (Q/D)x + a

=> C = (Q/D)x2/2 + ax + b

a) at x = 0, C = 5 x 1024 m-3

and x = L, C = 1.8 x 1024 m-3

=>. solving b = 5 x 1024 , a = [(-3.2 x 1024) - (Q/D)L2/2]/L => -9.73333 x 1027

=> C(x) = 6.666 x 1030 x2 - 9.73333 x 1027 x + 5 x 1024

b) at x = 0, C = 1.8 x 1024 m-3

and x = L, C = 1.8 x 1024 m-3

=>. solving b = 1.8 x 1024 , a = -(Q/D)L/2 => -3.3333 x 1027

=>   C(x) = 6.666 x 1030 x2 - 3.3333 x 1027 x + 1.8 x 1024

c) at x = 0, dC/dx = 0

and x = L, C = 1.8 x 1024 m-3

=>. solving , a= 0, b = 0.133333 x 1024

=>   C(x) = 6.666 x 1030 x2 + 1.3333 x 1023

b)  

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