UNIVERSITY OF ARIZONA
DEPARTMENT OF CHEMICAL AND ENVIRONMENTAL ENGINEERING
CHEE 502 - ADVANCED CHEMICAL ENGINEERING ANALYSIS
FALL 2008
Problem Set 8
1. Consider the spontaneous growth of algae in water, in the region close to the surface of a
stagnant pond. We will assume that the only factor limiting algal growth is light, and we will
also consider that diffusion of algae is very slow. Under these conditions, at any point in the
pond, the rate of growth of algae is a first order process with respect to algae concentration, with
a rate constant that is proportional to the intensity of light. A mass balance in algae leads to
bIN
t
N=
∂
∂
where N is algae concentration, I is intensity of light, b is a constant, and t is time measured so
that t=0 represents sunrise. If we assume that the most important factor that affects light
attenuation is the presence of algae, the change of light intensity with position is given by Beer's
law, with an extinction coefficient directly proportional to algae concentration,
aIN
z
I−=
∂
∂
where a is a constant and z represents depth in the pond. Initial and boundary conditions are
N=N
*
, t=0
where N
*
is a constant (i.e., algae concentration in the pond at sunrise is uniform); and
I=I
*
sin(
π
t/T), z=0
where T is the length of the day, and I
*
is the maximum incident light, which occurs at midday.
(a) Formulate the problem in terms of the following dimensionless variables:
T
t
=τ
x=aN
*
z
*
I
I
u
=
*
N
N
v
=
(b) Find u(
τ
,x) and v(
τ
,x) for the case in which algae growth is slow (
ε
=bI
*
T
→
0). Formulate
your results in terms of asymptotic expansions that include terms of O(
ε
1
).
