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ChE 452 Lecture 08
Analysis Of Data From A Batch Reactor
1
Objective
Data analysis from indirect measurements Essen’s method (learned in p-chem)
Does not usually work Van’t Hoff’s method
Accurate but amplifies errors in data
2
Background: Kinetic Data After Measuring
Indirect method – a method where you measure some other property (i.e. concentration vs time) and infer a rate equation. 3
0 5 10 15 20 25 30 3510
100
Time, Hours
Pre
ssur
e, x
2, to
rr
Figure 3.8 Typical batch data for reaction (3.7). Data of Tamaru[1955].
Objective For Today: Analysis Of Rate Data
Derive basic equations Essen’s method Van’t Hoff’s method
4
Derivation Of Performance Equation For A Batch
Reactor
5
For A B, the moles of A reacted/volume/time will equal the reaction rate, i.e.
A
A
dC=r
dτ(1)
CA is the concentration of A, is time, and rA is the rate of reaction per unit volume.
Figure 3.11 A batch reactor
Integration Yields The Following
6
(3.31)
Memorize this equation
0A
fA
C
A
AC
dC=τ
(-r )
For A First Order Reaction
rA = -k1CA
(3.38)
Substituting equation (3.38) into equation (3.31) and integrating yields:
7
fA
0A
1 C
CLn
k
1 Memorize this equation(3.39)
Derivation
For An nth Order Reaction:
nAnA Ckr
9
(3.41)
Substituting equation (3.41) into equation (3.31), integrating, and rearranging yields:
1C
C
Ck1n
11-n
fA
oA
1noAn
(3.42)
Memorize this equation
Derivation
Plots Of Equations
0% 20% 40% 60% 80% 100%
conversion
0
2
4
6
8
10
Tim
e
First Order
SecondOrder
11
Table 3.4 Rate Laws For A Number Of Reactions
12
Rate Laws for a number of reactions
Reaction Rate Law Differential Equation Integral Equation
A ProductsA+B Products
rA=kA
A ProductsA+B Products
rA=kA[A]
A ProductsA+B Products
rA=kA[A]n
A+B Products rA=kA[A][B]
A+2B Products
rA=kA[A][B]
A B rA=k1[A]-k2[B]
AA k
ddX
AAA Xk
ddX
nA
1nnAA
A X)C(kd
dX
)XCC)(X1(kd
dXA
0A
0BAAA
A
)XC2C)(X1(kd
dXAA
0BAA
A
A2AAA Xk)X1(k
d
dX
A
AX
k
A
A X11
Ln1
k
1X1
1
)C)(1n(
1k
1n
A1n0
AA
0AA
0B
A0B
0B
0A
ACXC
)X1(CLn
CC(
1k
0AA
0B
A0B
0B
0A
ACX2C
)X1(CLn
CC2(
1k
Ae
21 XX
1Ln
1)kk(
Fitting Batch Data To A Rate Law
Steps• Start with a batch reactor and
measure concentrations vs time.• Fit those data to a first order and a
second order rate law and see which equation fits better.
• Whichever rate equation fits best is assumed to be the correct rate equation for the reaction.
13
Key Challenge: First And Second Order Data Does Not Look That Much
Different
14
0 0.5 1 1.50
0.2
0.4
0.6
0.8
1
Time
C /
C0
AA
Second Order
0 0.5 1 1.50
0.2
0.4
0.6
0.8
1
Time
C /
C0
AA
Same k(CA
0)n-1
Vary k to fit data
Essen’s Method
15
fA
0A
1 C
CLn
k
1
1C
C
Ck1n
11-n
fA
oA
1noAn
First order
nth order
(3.42)
(3.39)
Essen’s Method
16
(C /
C )
- 1
0 AA
ln(C
/C
)0 A
A
0 0.5 1 1.50
0.2
0.4
0.6
0.8
1
TimeH
alf O
rder
Firs
t Ord
er
0 0.5 1 1.50
0.2
0.4
0.6
0.8
1
Time
Hal
f Ord
er
Figure 3.15
Example: The Concentration Of Dye As A Function Of
Time
17
CA,mmoles /Lit
, Min CA CA
1 0 0.63 6 0.45 120.91 1 0.59 7 0.43 130.83 2 0.56 8 0.42 140.77 3 0.53 9 0.40 150.71 4 0.50 10 0.38 160.67 5 0.48 11 0.37 17Table 3.5
Essen Plot For Example:
18
0 5 10 150
0.5
1
1.5
Time, Mins0 5 10 150
1
2
3
4
5
6
Time, Mins0 5 10 150
0.2
0.4
0.6
0.8
1
Time, Mins
ln(C
/C
)0 A
A
(C /
C )
- 1
2A
A
(C /
C )
- 1
0 AA
0
r2=.984 r2=.999r2=.981
No statistically significant difference between results.Figure 3.16
Example Shows Essen’s MethodDoes Not Distinguish Between
Models
In the literature, Essen’s method is often used.
Useful for impressing your boss since it always fits with good r2 (given good data)
It often gives the incorrect answers.
19
Van’t Hoff’s Method
• Take batch data as before.• Calculate kone (first order rate
constant) ktwo (second order rate).• kone should be constant for a first
order reaction, ktwo should be constant for a second order reaction. (Use f test to check).
20
Equations For kone And ktwo Follow From Before
21
A
0A
1 C
CLn
k
1
A
0A
1 C
CLn
1k
(3.39) (3.51)
1C
C
C)1n(
1k
1n
A
0A
1n0A
n
(3.52)
1C
C
Ck1n
11-n
fA
oA
1noAn
(3.42)
Solve for k1
Solve for kn
Derived previously
Derived previously
Easy Solution: Define A VB Module In Microsoft
Excel
22
Public Function kone(ca0, ca, tau) As Variantkone = Log(ca0 / ca) / tauEnd Function Public Function ktwo(ca0, ca, tau) As Variantktwo = ((1# / ca) - (1# / ca0)) / tauEnd Function Public Function kthree(ca0, ca, tau) As Variantkthree = ((1# / ca) ^ 2 - (1# / ca0) ^ 2) / tauEnd Function
Microsoft Excel/Visual Basic Return Types
23
As Variant General return type (can be an integer, real, vector, matrix, logical or text)
As Single Single precision real
As Double Double precision real
As Integer Integer
The Formulas In The Spreadsheet For Van’t
Hoff’s Method
24
B C D E F1 Ca0= 12 Essen's
Method3 time conc first second third4 ln(Ca0/Ca) (Ca0/Ca)-1 (CA0/CA)^
2-15 0 1 =kone(ca0,C5,B5) =ktwo(ca0,C5,B5) =kthree(ca0,C5,B5)
6 1 0.91 =kone(ca0,C6,B6) =ktwo(ca0,C6,B6) =kthree(ca0,C6,B6)
7 2 0.83 =kone(ca0,C7,B7) =ktwo(ca0,C7,B7) =kthree(ca0,C7,B7)
8 3 0.77 =kone(ca0,C8,B8) =ktwo(ca0,C8,B8) =kthree(ca0,C8,B8)
9 4 0.71 =kone(ca0,C9,B9) =ktwo(ca0,C9,B9) =kthree(ca0,C9,B9)
The Numerical Values For
Van’t Hoff’s Method
25
B C D E F3 Time conc k1 k2 k3
4 0 1 ln(1/Ca)/t
((Ca0/Ca)-1)/t
((CA0/CA)^2-1)/t/2
5 1 0.91 0.094 0.099 0.1046 2 0.83 0.093 0.102 0.1137 3 0.77 0.087 0.1 0.1148 4 0.71 0.086 0.102 0.1239 5 0.67 0.08 0.099 0.123
10 6 0.63 0.077 0.098 0.127
Van’t Hoff Plot
26
Van’t Hoff’s method is much more accurate than Essens’ method.Essen’s is more common!
0 5 10 150.05
0.1
0.15
0.2
Time, Mins
Rat
e C
onst
ant
K3
K2
K1
Oxidation ofRed Dye
Figure 3.18 Van’t Hoff plot of the data from tables 3.5 and 3.6
Discussion Problem: Use Van’t Hoff’s Method To Determine The Order For
The Following Data
27
Table 4.1 Buchanan’s [1871] data for the reaction:
CICH2COOH + H2COOH + HCI at 100º C
Time Hours [CICH2COOH] gms/liter
02346
10131928
34.54348
43.803.693.603.473.102.912.542.261.951.591.39
Solution:
28
Ca0= 4Van't Hoff's
time Conc first second thirdln(ca0/Ca) (Ca0/Ca)-1 (CA0/CA)̂ 2-
10 4 =kone(cao,B5
,A5)=ktwo(cao,B5,A5)
=kthree(cao,B5,A5)
2 3.8 =kone(cao,B6,A6)
=ktwo(cao,B6,A6)
=kthree(cao,B6,A6)
3 3.69 =kone(cao,B7,A7)
=ktwo(cao,B7,A7)
=kthree(cao,B7,A7)
4 3.6 =kone(cao,B8,A8)
=ktwo(cao,B8,A8)
=kthree(cao,B8,A8)
6 3.47 =kone(cao,B9,A9)
=ktwo(cao,B9,A9)
=kthree(cao,B9,A9)
Solution Continued:
29
ca0= 4
time conc first second third
ln(ca0/Ca)
(Ca0/Ca)-1
(CA0/CA)^2-1
0 4 #VALUE! #VALUE! #VALUE!2 3.8 0.026 0.007 0.0033 3.69 0.027 0.007 0.0044 3.6 0.026 0.007 0.0046 3.47 0.024 0.006 0.00310 3.1 0.025 0.007 0.00413 2.91 0.024 0.007 0.00419 2.54 0.024 0.008 0.00525 2.26 0.023 0.008 0.005
34.5 1.95 0.021 0.008 0.00643 1.59 0.021 0.009 0.00848 1.39 0.022 0.01 0.009
Van't Hoff's
Van’t Hoff Plot
30
0 10 20 30 40 500.02
0.03
0.04
0.05
Time, Mins
Rat
e C
onst
ant
K3
K2
K1
Hydration ofChloracetic Acid
Figure 3.18 Van’t Hoff plot of the data from tables 3.5 and 3.6
Discussion Problem 2
Ammonium-dinitramide, (ADN) NH4N(NO2)2, is a oxidant used in solid fuel rockets and plastic explosives. ADN is difficult to process because it can blow up. Oxley et. Al., J. Phys chem A, 101 (1997) 5646, examined the decomposition of ADN to try to understand the kinetics of the explosion process. At 160º C they obtained the data in Table P3.20.
31
time, seconds
fraction of the AND remaining
time, seconds
fraction of the AND remaining
time, seconds
fraction of the AND remaining
0 1.0 900 0.58 2400 0.24
300 0.84 1200 0.49
600 0.70 1500 0.41
Table P3.20 Oxley's measurements of the decomposition of dinitramide at 160 C
Discussion Problem 2 Continued:
a) Is this a direct or indirect measurement of the
rate? b) Use Van’t Hoff’s Method to fit this data to a rate
equation. c) If you had to process ADN at 160° C, how long
could you run the process without blowing anything up? Assume that there is an explosion hazard once 5% of the ADN has reacted to form unstable intermediates.
32
This Is An Indirect Measurement! Use Same Spreadsheet As Before
To Fit Data
33
Ca0= =b5
Van't Hoff's
time Conc first second third
0 1 ln(ca0/Ca) (Ca0/Ca)-1 (CA0/CA)^2-1
300 0.84 =kone(cao,B5,A5)
=ktwo(cao,B5,A5)
=kthree(cao,B5,A5)
600 0.7 =kone(cao,B6,A6)
=ktwo(cao,B6,A6)
=kthree(cao,B6,A6)
900 0.58 =kone(cao,B7,A7)
=ktwo(cao,B7,A7)
=kthree(cao,B7,A7)
1200 0.49 =kone(cao,B8,A8)
=ktwo(cao,B8,A8)
=kthree(cao,B8,A8)
1500 0.41 =kone(cao,B9,A9)
=ktwo(cao,B9,A9)
=kthree(cao,B9,A9)
2400 0.24 =kone(cao,B10,A10)
=ktwo(cao,B10,A10)
=kthree(cao,B10,A10)
Solution Cont.
34
Ca0= 1
Van't Hoff's time Conc first second third
ln(ca0/Ca) (Ca0/Ca)-1 (CA0/CA)^2-1
0 1 #VALUE! #VALUE! #VALUE!
300 0.84 0.000581 0.000635 0.001391
600 0.7 0.000594 0.000714 0.001735
900 0.58 0.000605 0.000805 0.002192
1200 0.49 0.000594 0.000867 0.002637
1500 0.41 0.000594 0.000959 0.003299
2400 0.24 0.000595 0.001319 0.006817
Solution Cont.
c) from equ 3.39
35
fA
oA
1 C
CLn
k
1
kk1=0.0006/sec (from spreadsheet)
oAC =1 (given) fAC =0.95 (what's left if 5% converted)
sec850.95
1Ln
0.0006
1
C
CLn
k
1fA
oA
1
Summary: Two Methods To Fit Rate Data
• Essen’s Method• Most common method• Plots look the best• Gives great looking results even with incorrect
rate equation • Van’t Hoff’s Method
• More accurate than Essen• Rare in literature• Plots noisier• Highlights weaknesses in rate equations
36
Class Question
What did you learn new today?
37