This preview shows pages 1–2. Sign up to view the full content.
CHEE 3334
Fall ’99
Michael Nikolaou
FINAL EXAM
Openbook, opennotes.
Total points 100.
Please, budget your time
!
Do NOT try to read the textbook during
the exam!
Write neatly!
Do NOT use your programmable calculator to find results you are asked to compute
in a certain wayafter all, the calculator may find the wrong answer or not all answers!
Show all work.
Return this sheet with your exam
.
GOOD LUCK!
(70 pts.)
1.
A continuousflow industrial fermentation reactor was initially at steady state.
Then, an engineer conducted
two experiments to determine how the reactor yield deviates from its steady state value as a result of the
flowrate to the reactor deviating from its steady state.
In each experiment she changed, over time, the
flowrate from its steady state value, and measured how the reactor yield responded to that change.
The
results of her experiments are shown in the following two Figures and corresponding Tables, where
k
is
time in hours,
u
(
k
) is the difference between the changed flowrate value and its steadystate value, and
y
(
k
)
is the difference between the resulting yield value and its steadystate value, in dimensionless units.
0 .0 0
0 .5 0
1 .0 0
1 .5 0
2 .0 0
0
2
4
6
8
1 0
 2 .0 0
 1 .0 0
0 .0 0
1 .0 0
2 .0 0
0
2
4
6
8
1 0
Table 1
k
u
(
k
)
y
(
k
)
0
1
0.00
1
1
1.39
2
1
1.53
3
1
0.95
4
1
1.42
5
1
1.00
6
1
0.69
7
1
1.50
8
1
1.36
9
1
1.39
10
1
0.89
Table 2
k
u
(
k
)
y
(
k
)
0
1
0.00
1
1
1.54
2
1
0.75
3
1
0.99
4
1
1.14
5
1
0.85
6
1
1.45
7
1
0.73
8
1
1.54
9
1
1.21
10
1
0.65
The engineer postulated that the dynamic model describing the relationship between
u
and
y
is
)
2
(
)
1
(
)
(

+

=
k
bu
k
au
k
y
(1)
and suggested that the leastsquares method applied to errors from
2
=
k
to
10
=
k
should be used to
estimate the unknown parameters.
a.
On the basis of the data in Table 1
only
, estimate the parameters
a
and
b
by using the linear leastsquares
method.
Are your estimates unique?
{
{
1.53
1 1
(2)
(1)
(0)
0.95
1 1
(3)
(2)
(1)
0.89
1 1
(10)
(9)
(8)
y
y
X
X
y
u
u
a
a
y
u
u
e
b
b
y
u
u
θ
=

=

M
M M
M
M
M
14243
123
142 43
1 442 4 43
The optimal values of the parameters a, b are solutions of the equations
ˆ
9
9
10.73
ˆ
ˆ
9
9
10.73
T
T
a
X X
X y
b
=
=
.
Gaussian elimination yields:
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
 Fall '06
 nikolau

Click to edit the document details