Analysis of Data From a Batch Reactor

7/31/2019 Analysis of Data From a Batch Reactor

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Analysis Of Data From A Batch Reactor

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Objective

Data analysis from indirect measurements

Essens method (learned in p-chem)

Does not usually work

Vant Hoffs method

Accurate but amplifies errors in data

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Background: Kinetic Data AfterMeasuring

Indirect method a method where you measuresome other property (i.e. concentration vs time)

and infer a rate equation. 3

0 5 10 15 20 25 30 35

10

100

Time, Hours

Pressu

re,x2,

torr

Figure 3.8 Typical batch datafor reaction (3.7). Data of

Tamaru[1955].


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Objective For Today:Analysis Of Rate Data

Derive basic equations

Essens method

Vant Hoffs method

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Derivation Of PerformanceEquation For A Batch Reactor

5

For AB, the moles of Areacted/volume/time will equal the reactionrate, i.e.

A

A

dC =rd

(1)

CA is the concentration of A, is time,and rA is the rate of reaction per unitvolume. Figure 3.11 A batch reactor


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Integration Yields TheFollowing

6

(3.31)

Memorize this equation

0A

fA

C

A

AC

dC=

(-r )


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For A First Order Reaction

rA

= -k1

CA

(3.38)

Substituting equation (3.38) into equation

(3.31) and integrating yields:

7

f

A

0

A

1 C

CLn

k

1 Memorize this equation(3.39)

Derivation


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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

1no

An

(3.42)

Memorize this equation

Derivation


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Plots Of Equations

0% 20% 40% 60% 80% 100%

conversion

0

2

4

6

8

1 0

Time

First Order

Second

Order

11

bl 3


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Table 3.4 Rate Laws For ANumber Of Reactions

12

Rate Laws for a number of reactions

Reaction Rate Law Differential Equation Integral Equation

A Products

A+B Products

rA=kA

A ProductsA+B Products

rA=kA[A]

A Products

A+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

d

dX

AAA Xk

ddX

nA

1nnAA

A X)C(kd

dX

)XCC)(X1(kd

dXA

0A

0BAAAA

)XC2C)(X1(kd

dXAA

0BAA

A

A2AAA Xk)X1(k

d

dX

AA

Xk

AA

X11Ln1k

1

X1

1

)C)(1n(

1k

1n

A1n0

A

A

0AA0B

A

0

B0B

0A

ACXC

)X1(CLn

CC(

1k

0AA

0B

A0B

0B

0A

ACX2C

)X1(CLn

CC2(

1k

Ae21

XX

1Ln

1)kk(

Fi i B h D T A R


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Fitting Batch Data To A RateLaw

Steps

Start with a batch reactor and measureconcentrations vs time.

Fit those data to a first order and a

second order rate law and see whichequation fits better.

Whichever rate equation fits best is

assumed to be the correct rate equationfor the reaction.

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Key Challenge: First And Second OrderData Does Not Look That Much Different

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0 0 . 5 1 1 .

0

0 . 2

0 . 4

0 . 6

0 . 8

1

Time

C

/C

0

A

A

HalfOrder

FirstOrder

Second Orde

ThirdOrder

0 0 . 5 1 1 .

0

0 . 2

0 . 4

0 . 6

0 . 8

1

Time

C

/C0

A

A

HalfO

rder

FirstOrder

SecondOrder

ThirdO

rder

Samek(CA

0)n-1

Vary k to fitdata


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Essens Method

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f

A

0

A

1 C

CLn

k

1

1C

C

Ck1n

1 1-n

fA

oA

1noAn

First order

nth order

(3.42)

(3.39)


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Essens Method

16

(C

/C)

-1

0A

A

ln(C

/C

)

0A

A

0 0 . 5 1 1 .

0

0 . 2

0 . 4

0 . 6

0 . 8

1

Time

Ha

lfO

rder

Fir

stO

rder

Second

Order

Third

Order

0 0 . 5 1 1 .

0

0 . 2

0 . 4

0 . 6

0 . 8

1

Time

Half

Ord

er

Fir

stOrder

Second

Order

Third

Order

Figure 3.15

E l Th C t ti Of


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Example: The Concentration OfDye As A Function Of Time

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CA,mmoles

/Lit

, Min CA CA

1 0 0.63 6 0.45 120.91 1 0.59 7 0.43 13

0.83 2 0.56 8 0.42 14

0.77 3 0.53 9 0.40 150.71 4 0.50 10 0.38 16

0.67 5 0.48 11 0.37 17

Table 3.5


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Essen Plot For Example:

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0 5 10 15

0

0.5

1

1.5

Time, Mins0 5 10 15

0

1

2

3

4

5

6

Time, Mins0 5 10 15

0

0.2

0.4

0.6

0.8

1

Time, Mins

ln(C/

C

)

0A

A

(C

/C

)

-1

2

A

A

(C

/C)

-1

0A

A

0

r2=.984 r2=.999r2=.981

No statistically significant difference between results.

Figure 3.16

E l Sh E M th d


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Example Shows Essens Method

Does Not Distinguish Between Models

In the literature, Essens method is often used.

Useful for impressing your boss since it alwaysfits with good r2 (given good data)

It often gives the incorrect answers.

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Vant Hoffs Method

Take batch data as before.

Calculate kone (first order rate constant)ktwo (second order rate).

kone should be constant for a first orderreaction, ktwo should be constant for asecond order reaction. (Use f test tocheck).

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Equations For kone And


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Equations For kone Andktwo Follow From Before

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A

0

A

1 C

CLn

k

1

A

0

A1

C

CLn

1k

(3.39) (3.51)

1C

C

C)1n(

1k

1n

A

0A

1n0A

n

(3.52)

1C

C

Ck1n

1 1-n

fA

oA

1noAn

(3.42)

Solve for k1

Solve for kn

Derived previously

Derived previously

Easy Solution: Define A VB


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Easy Solution: Define A VBModule In Microsoft Excel

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Public Function kone(ca0, ca, tau) As Variant

kone = Log(ca0 / ca) / tau

End Function

Public Function ktwo(ca0, ca, tau) As Variant

ktwo = ((1# / ca) - (1# / ca0)) / tau

End Function

Public Function kthree(ca0, ca, tau) As Variant

kthree = ((1# / ca) ^ 2 - (1# / ca0) ^ 2) / tau

End Function

Microsoft Excel/Visual Basic


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Microsoft Excel/Visual BasicReturn Types

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As Variant General return type (can be aninteger, real, vector, matrix,logical or text)

As Single Single precision real

As Double Double precision real

As Integer Integer


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Spreadsheet For Vant Hoffs

Method

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B C D E F1 Ca0= 1

2 Essen's

Method

3 time conc first second third4 ln(Ca0/Ca) (Ca0/Ca)-1 (CA0/CA)^

2-1

5 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)



T e Numer ca Va ues For


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T e Numer ca Va ues ForVant Hoffs Method

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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 0.104

6 2 0.83 0.093 0.102 0.113

7 3 0.77 0.087 0.1 0.1148 4 0.71 0.086 0.102 0.123

9 5 0.67 0.08 0.099 0.123

10 6 0.63 0.077 0.098 0.127


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Vant Hoff Plot

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Vant Hoffs method is much more accurate than Essens method.

Essens is more common!

0 5 10 150.05

0.1

0.15

0.2

Time, Mins

RateC

onstant

K3

K2

K1

Oxidation ofRed Dye

Figure 3.18Vant Hoff plot of the data from tables 3.5 and 3.6

scuss on ro em: se an o s


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scuss on ro em: se an o sMethod To Determine The Order For The

Following Data

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Table 4.1 Buchanans [1871] data for the reaction:

CICH2COOH + H2COOH + HCI at 100 C

Time Hours [CICH2COOH] gms/liter

0

2

3

4

6

10

1319

28

34.5

43

48

4

3.80

3.69

3.60

3.47

3.10

2.912.54

2.26

1.95

1.59

1.39


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Solution:

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Ca0= 4

Van't Hoff's

time Conc first second third

ln(ca0/Ca) (Ca0/Ca)-1 (CA0/CA)^2-

0 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)


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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 0.003

3 3.69 0.027 0.007 0.004

4 3.6 0.026 0.007 0.004

6 3.47 0.024 0.006 0.003

10 3.1 0.025 0.007 0.004

13 2.91 0.024 0.007 0.004

19 2.54 0.024 0.008 0.005

25 2.26 0.023 0.008 0.005

34.5 1.95 0.021 0.008 0.006

43 1.59 0.021 0.009 0.008

48 1.39 0.022 0.01 0.009

Van't Hoff's


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Vant Hoff Plot

30

0 10 20 30 40 500.02

0.03

0.04

0.05

Time, Mins

RateConstant

K3

K2

K1

Hydration of

Chloracetic Acid

Figure 3.18Vant Hoff plot of the data from tables 3.5 and 3.6


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Discussion Problem 2

Ammonium-dinitramide, (ADN) NH4N(NO2)2, is a oxidant used in solidfuel rockets and plastic explosives. ADN is difficult to process because itcan blow up. Oxley et. Al., J. Phys chem A, 101 (1997) 5646, examinedthe decomposition of ADN to try to understand the kinetics of theexplosion process. At 160 C they obtained the data in Table P3.20.

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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 ofdinitramide at 160 C

Discussion Problem 2


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Discussion Problem 2Continued:

a) Is this a direct or indirect measurement of the rate?

b) Use Vant Hoffs Method to fit this data to a rateequation.

c) If you had to process ADN at 160 C, how long couldyou run the process without blowing anything up?Assume that there is an explosion hazard once 5% ofthe ADN has reacted to form unstable intermediates.

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Same Spreadsheet As Before To Fit

Data

33

Ca0= =b5

Van't Hoff's


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,B1

0,A10)

=ktwo(cao,B1

0,A10)

=kthree(cao,

B10,A10)


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Solution Cont.

34

Ca0= 1

Van't Hoff's


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


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Solution Cont.

c) from equ 3.39

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fA

oA

1 C

CLn

k

1

k

k1=0.0006/sec (from spreadsheet)o

AC =1 (given)f

AC =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


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Summary: Two Methods To FitRate Data

Essens Method

Most common method

Plots look the best

Gives great looking results even with incorrect rateequation

Vant Hoffs Method

More accurate than Essen

Rare in literature

Plots noisier Highlights weaknesses in rate equations

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Class Question

What did you learn new today?

Analysis of Data From a Batch Reactor

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