10.37 Chemical and Biological Reaction Engineering, Spring 2007
Prof. William H. Green
Lecture 12: Data collection and analysis
This lecture covers: Experimental methods for the determination of kinetic
parameters of chemical and enzymatic reactions; determination of cell growth
parameters; statistical analysis and model discrimination
Continuing the stability and multiple steady-state discussion from Lecture 11:
removal of heat
R (t )
•
G (t )
•
Three steady-state solutions
generation
•
Treactor
Figure 1. Three steady-state conditions shown on a G(T) versus T graph.
⎛ ξ1 ⎞
⎜ ⎟
⎜ ξ 2 ⎟ = z SS
⎜T ⎟
⎝ ⎠ SS
dξ1
=0
dt
dξ 2
=0
dt
dT
=0
dt
steady-state
⎧ dξ1
⎫
⎪ dt = f1 (ξ1 , ξ 2 , T ) ⎪
⎪
⎪
⎪ dξ 2
⎪ dz
original eqns. ⎨
= f 2 (ξ1 , ξ 2 ,T ) ⎬ →
= F ( z)
dt
dt
⎪
⎪
⎪ dT
⎪
= f 3 (ξ1 , ξ 2 , T ) ⎪
⎪
⎭
⎩ dt
stability: we want any perturbation
δz
from
vector notation
z SS to be self correcting
Cite as: William Green, Jr., course materials for 10.37 Chemical and Biological Reaction Engineering,
Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology.
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i.e.
d
(δ z ) = (−ve)δ z
dt
δz
what does perturbation cause?
- back to steady-state or off elsewhere?
•
Figure 2. A small perturbation moves the system away from steady state. Does the
system move back or does it move to elsewhere?
dz
= F( z)
dt
#
z = z SS + δ z
δ z = z − z SS
d
dz
(δ z ) =
= F (δ z + z SS )
dt
dt
0
0
dF
2
δ z + F ( z SS ) + O(δ z )
≈
dz
Jacobian matrix
⎛ dF ⎞
d
(δ z ) = ∑ ⎜ n ⎟ δ zm
dt
⎝ dzm ⎠ z
SS
= Jδz
⎛ df1
⎜
⎜ dξ1
⎜ df
J =⎜ 2
⎜ dξ1
Jacobian ⎜ df
⎜ 3
⎝ dξ1
df1
dξ 2
df 2
dξ 2
df 3
dξ 2
df1 ⎞
⎟
dT ⎟
df 2 ⎟
⎟
dT ⎟
df 3 ⎟
⎟
dT ⎠
Matrix
d
(δ z ) = M (δ z )
dt
if eigenvalues of M<0 then stable
10.37 Chemical and Biological Reaction Engineering, Spring 2007
Prof. William H. Green
Lecture 12
Page 2 of 4
Cite as: William Green, Jr., course materials for 10.37 Chemical and Biological Reaction Engineering,
Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology.
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CO + H 2O R CO2 + H 2 + Q
toxic
heat
1
doesn’t occur if equilibrium limited
w/ catalyst
X
equilibrium
original
rxn
T
Figure 3. Conversion (X) versus Temperature (T).
Data Collection:
-
determining rate laws
Products,
Unreacted Stuff,
Byproducts
Reactor
Reactants
CoT ,τ
Figure 4. Schematic of a general reactor.
make plots
- get empirical expression
Æ fit curve
product
conc.
at
output
[ A]0
Figure 5. Product concentration versus reactant concentration A.
r(conc,T)
ouput-input
10.37 Chemical and Biological Reaction Engineering, Spring 2007
Prof. William H. Green
Lecture 12
Page 3 of 4
Cite as: William Green, Jr., course materials for 10.37 Chemical and Biological Reaction Engineering,
Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology.
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→ rV
∫ r dxdydz
if homogeneous
dV
“well-stirred” reactor
(slow reactions)
“no” conversion
(really ~.1% conversion)
C = C0 ± .1% Æ can measure (output-input)
(r barely changes)
*need very sensitive product detection
“differential reactor”
From data:
guess mechanism
vary ( k , k eq )Æmake a fit
1) Is mechanism consistent (error bars?) w/ data?
2) How to regress k ? (least squares method)
10.37 Chemical and Biological Reaction Engineering, Spring 2007
Prof. William H. Green
Lecture 12
Page 4 of 4
Cite as: William Green, Jr., course materials for 10.37 Chemical and Biological Reaction Engineering,
Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology.
Downloaded on [DD Month YYYY].