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CHEE 3334
Fall 2003
Michael Nikolaou
Student Name
Final Examination
Openeverything.
Total points 100.
Please, budget your time
!
Do NOT try to read the book during the exam!
Write
neatly!
Do NOT use your programmable calculator to find results you are asked to compute in a certain way (but you
may use anything you want to check numerical results)!
PLEASE SHOW ALL RELEVANT WORK IN THE INDICATED SPACES.
DO NOT
RETURN ADDITIONAL PAGES.
(You may use scratch paper only for scratch work.)
1.
(40 pts.)
Continuous biochemical reactors, such as the one shown on the right, are
used in a variety of processes, from waste treatment, to alcohol fermentation, to
production of antibiotics.
In a biochemical reactor, cells (biomass) consume
nutrients (usually called
substrate
) such as glucose, to grow, multiply, and produce
useful chemicals after consuming the substrate.
A component mass balance on a
continuousflow biochemical reactor (frequently called a
chemostat
) yields the
ordinary differential equations
(
29
max 2
1
1
2
max 2 1
2
2
2
2
(
)
m
f
m
x
dx
D x
dt
k
x
x x
dx
D x
x
dt
Y k
x
μ
=

+
=


+
where
1
x
=
mass of cells per unit volume,
2
x
=
mass of substrate per unit volume.
The parameters appearing in the model are
1
max
0.53
hr

=
,
0.12
/
m
k
g liter
=
,
0.4
Y
=
(yield),
1
0.3
D
hr

=
(dilution rate = volumetric flow rate / volume =
F
/
V
), and
2
4
/
f
x
g liter
=
(concentration of
substrate in the feed stream).
The initial conditions for this reactor are
1
(0)
1.5
/
x
g liter
=
and
2
(0)
0.5
/
x
g liter
=
a.
Are the above differential equations linear or nonlinear?
(Explain.)
The equations can be written as
(
29


=
+


+
=
+
max 2
1
1
2
max 2 1
2
2
2
2
0
0
(
)
m
f
m
x
dx
D x
dt
k
x
x x
dx
D x
x
dt
Y k
x
or
( )
0
L x
=
where
1
2
x
x
x
=
Clearly,
(
)
( )
( )
L x
y
L x
L y
+
+
, i.e.
+
1
+
+
1
1
1
1
2
2
2
2
(
)
(
)
(
)
x
y
x
y
L
L
L
x
y
x
y
.
Therefore, the equations are nonlinear.
b.
Perform one iteration of Euler’s method with time step
0.1
t
∆ =
, to compute
(
29
1
1
x t
and
(
29
2
1
x
t
.
Euler’s method:
 page 1 of 7 
F, x
2
f
F, x
2
,
x
1
V
k
tk
x1(tk)
x2(tk)
mumax = 0.53
0
0
1.5
0.5
km = 0.12
1
0.1
1.519113
0.444718
Y = 0.4
2
0.2
1.536944
0.392865
D = 0.3
3
0.3
1.553234
0.345083
x2f = 4
4
0.4
1.567718
0.302028
dt = 0.1
0
0.5
1
2
0
1
2
3
4
5
6
x1
x2
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29
μ
+
+
=
+ ∆

+
=
+ ∆


+
max 2
1
1
1
1
2
max 2
1
2
2
2
1
2
2
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
(
))
k
k
k
k
m
k
k
k
k
k
k
f
m
k
x
t
x t
x t
t
D x t
k
x
t
x
t
x t
x
t
x
t
t D x
x
t
Y k
x
t
c.
Substitute the derivatives
1
dx
dt
and
2
dx
dt
at time
t
k
by
backward
finite divided differences.
What numerical method
would you use to calculate
(
29
1
k
x t
and
(
29
2
k
x
t
in terms of known past values
(
29
1
1
k
x t

and
(
29
2
1
k
x
t

?
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This note was uploaded on 04/15/2008 for the course CHEE 3334 taught by Professor Nikolau during the Fall '06 term at University of Houston.
 Fall '06
 nikolau

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