AN ABSTRACT OF THE THESIS OF Robert Allen Knudsen for the M.S. in Chemical Engineering (Name) (Degree) (Major) Date thesis is presented Title CHEMICAL REACTION STUDIES BY FREQUENCY RESPONSE METHODS Abstract approved Major pro sor A method for measuring chemical reaction rates in order to distinguish between different reaction mechanisms is presented. This method was used to study zero -, first- and second -order re- actions by analyzing the output response to a sinusoidal input of reactant concentrations in an isothermal continuous stirred tank reactor. The results showed that only zero- and first -order reactions produce a pure sinusoidal output, and that the harmonics of a modi- fied Fourier series are less than ten percent of the fundamental for a second -order reaction. A complex rate expression was also investigated, but the trial and error technique performed on an IBM 7090 digital computer did not converge. 3,1969
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AN ABSTRACT OF THE THESIS OF
Robert Allen Knudsen for the M.S. in Chemical Engineering (Name) (Degree) (Major)
Date thesis is presented
Title CHEMICAL REACTION STUDIES BY FREQUENCY
RESPONSE METHODS
Abstract approved Major pro sor
A method for measuring chemical reaction rates in order to
distinguish between different reaction mechanisms is presented.
This method was used to study zero -, first- and second -order re-
actions by analyzing the output response to a sinusoidal input of
reactant concentrations in an isothermal continuous stirred tank
reactor.
The results showed that only zero- and first -order reactions
produce a pure sinusoidal output, and that the harmonics of a modi-
fied Fourier series are less than ten percent of the fundamental for
a second -order reaction.
A complex rate expression was also investigated, but the trial
and error technique performed on an IBM 7090 digital computer did
not converge.
3,1969
If more theoretical responses are computed for other, and
more complex, rate equations, this approach can be used to distin-
guish between different reaction rate equations and consequently can
aid in determining the true reaction mechanism.
CHEMICAL REACTION STUDIES BY FREQUENCY RESPONSE METHODS
by
ROBERT ALLEN KNUDSEN
A THESIS
submitted to
OREGON STATE UNIVERSITY
in partial fulfillment of the requirements for the
degree of
MASTER OF SCIENCE
June 1964
APPROVED:
Assistant Professor Chemical Engineering
In Charge of Major
Head of Department of Chemical Engineering
Dean of Graduate School
Date thesis is presented 31, /969
ACKNOWLEDGEMENTS
The author wishes to express his appreciation to Professor
D. E. Jost for his help and encouragement during this investigation
and also his suggestions for writing the thesis. Dr. H. E. Goheen
is to be thanked for introducing me to numerical calculus, and
Dr. R. E. Gaskell deserves credit for introducing the Fourier
series solution.
The possibilities of analog computers were discovered through
limited experimental investigations performed by this author and
Professors W. W. Smith and S. A. Stone.
I am also indebted to my friend, Mr. James R. Divine, for
his help during this work.
Finally, the Engineering Experiment Station is to be thanked
for their generous financial support.
TABLE OF CONTENTS
Pag e
INTRODUCTION 1
REVIEW OF LITERATURE 3
THEORY 5
CALCULATIONAL METHODS 12
Numerical Calculus Method Analytic Steady -State Solutions Fourier Series Solution Analog Computer Solution
DISCUSSION OF RESULTS
12 12 13 15
16
Nonreacting System 16 Zero -Order Reactions 16 First -Order Reaction 17 Second -Order Reaction 19 Complex Reaction 25 Numerical Calculus Method 27 Experimental Considerations 27
CONCLUSIONS AND RECOMMENDATIONS 30
BIBLIOGRAPHY 31
NOMENCLATURE 33
APPENDICES
Appendix A Digital Computer Programs 36
Numerical Calculus Scheme 36 Fourier Series Solution of Eq. (5) for a 38
Second -Order Reaction Fourier Series Solution of Eq. (5) for a 41
Complex Rate Equation
Appendix B Analog Computer Programs
Rate Equation Not Influenced by Products Rate Equation Influenced by Products
Pag e
52
52 52
Appendix C Digital Computer Data for Second- 53 Order Reaction
LIST OF TABLES
Table Page
1 Periodic steady -state constants in the Foirier series solution for Eq. (5) when f(n) = Krk
and A = 0.5
53
2 Periodic steady -state constants in the Forier 54 series solution for Eq. (5) when f(ri) = Krk
and K = 1.0
LIST OF FIGURES
Figure Page
1 Reactor model 6
2 Amplitude ratio and phase lag for zero-order 17and nonreacting systems
3 Amplitude ratio and phase lag for first-order 18reactions
4 Mean dimensionless output concentration for 20a second-order reaction for A = 0. 5
5 Amplitude ratio and phase lag of the fundamental 21for a second-order reaction when A = 0. 5
6 Amplitude ratio and phase lag of the second har- 22monic for a second-order reaction when A = 0. 5
7 Amplitude ratio and phase lag of the second har- 23monic for a second-order reaction when A = 0. 5
8 Mean dimensionless output for various input am- 24plitudes and a second-order reaction when K= 1. 0
9 Amplitude ratios with respect to A of fundamental 25and first harmonic to fundamental for a second-
order reaction when K = 1„0
10 Dimensionless response curves for a first- and 28second-order reactions when A=0. 1, K = 1. 0
and Q = 10
d'n11 Block diagram of analog computer for -sr =1 52
+ A sinf2G- r\ - f(n)
12 Block diagram of analog computer for—H =1 52+ A sinflG-'n - f(T),v) d0
CHEMICAL REACTION STUDIES BY FREQUENCY RESPONSE METHODS
INTRODUCTION
For a given chemical system undergoing reaction, determina-
tion of the form of the rate equation and of the constants appearing in
the equation is required for optimum design of a chemical reactor.
Knowledge of the rate equation is also a first step toward determining
the actual reaction mechanism. Once a reaction mechanism is estab-
lished, extrapolation to other concentrations and temperatures may be
achieved with a greater degree of assurance, and thus, optimum con-
ditions for conducting the reaction can be established. Therefore, a
primary aim of applied kinetic studies is to determine the rate equa-
tion, and if possible, to determine the reaction mechanism.
In general, the form of the rate equation cannot be predicted
from the stoichiometric equation, and a trial and error procedure is
required where experimental rate data are plotted to test various
postulated rate expressions. As a starting point, simple rate equa-
tions are tested; however, equations of simple form are often not
sufficient since the actual mechanism is usually complex and involves
free radicals, ions or polar substances, molecules, or other inter-
mediate species. The postulation of a particular complex reaction
mechanism from a large number of possibilities is not straight
2
forward; the choice usually being based on an intimate knowledge
of the chemistry of similar reactions, fragmentary experimental
data, and intuition.
Sometimes two or more mechanisms can be consistent with
the available rate and thermodynamic data. In these cases, speci-
fic tests, e. g. , initial rate experiments, are devised to disprove
all but one of the postulated mechanisms.
The approach presented here provides a more definitive
characterization of the chemical rate process by measuring the
frequency- response of a reacting system over a wide range of per-
turbation frequencies, input concentration amplitudes, and tempera-
ture. Theoretically, no two different rate equations will exhibit the
same response, and, therefore, this approach can distinguish be-
tween different expressions for the reaction rate and, consequently
between postulated mechanisms.
3
LITERATURE REVIEW
In the early thirties, H. S. Black (1), an electrical engineer,
used the results of H. Nyquist's (6) work on the stability of feed
back amplifiers by frequency- response analysis to investigate nega-
tive feedback by this same technique. His investigation, which was
the first major application of frequency- response, made possible
the development of the transcontinental telephone as well as modern
radio and television (7 p. v). Since the pioneering work of Black, much
has been published on frequency- response methods in the fields of
electrical engineering and process control, but only a few articles
have been concerned with the application of these methods to systems
in which a chemical reaction occurs.
The principal application of frequency- response analysis in
chemical engineering has been in process control. In this connec-
tion, Bilous, Black and Piret (2) related this method to the control
of chemical reactors.
Perturbation methods, other than frequency- response, have
been used to investigate chemical kinetics. A theoretical study of
the effects of simultaneous periodic variations in temperature and
volume for chain reactions was made by H. M. Wight (10). The
main interest was to investigate the frequency dependence of the
rate enhancement for chain reactions. A Swedish patent was granted
4
to P. 0. Stelling and R. B. M. Elkund (8) for the acceleration of
chemical reactions by vibration. Perturbation methods were utilized
by M. Eigen (3) to study the kinetics of extremely fast ionic reactions
in aqueous solutions. The kinetics of nitrogen oxidation in an oscil -
lating discharge have been examined by S. S. Vasil'ev and M. S.
Selivokhina (9), and F. A. Williams (11) investigated the response
of a burning solid to small - amplitude pressure oscillations.
The above mentioned perturbation studies have all considered
only small perturbations so as to eliminate the generation of higher
harmonics and to simplify the analysis. This investigation will
show that a more complete characterization of the rate process can
be established by considering the full range of perturbations.
5
THEORY
The theoretical approach taken here will help distinguish
between various forms of the rate equation by determining different
responses to sinusoidal input pulses of reactant concentrations in an
isothermal continuously stirred -tank reactor. One complex and three
simple rate expressions were selected for study. The complex rate
expression
KiCD CE gD-
1 + K2CD (1)
was taken from a recent applied kinetics book (5, p. 20 -22). In Eq. (1)
Fig. 10. Dimensionless response curves for a first- and second -order reactions when A = 0. 1,
K= 1.0 and S2= 10
ó 0.5'7 u cd a) Ir
0 0.5a-
0
w
a °
0.6
0.6 m
0
first -order
29
analytical techniques, the conduction of experiments as suggested
here is not unreasonable.
The following stipulations must be met:
(1) Input concentration of the reactants must vary in a sinusoidal manner.
(2) Concentration in the reactor must be uniform.
(3) Temperature of the reactor must be constant.
(4) Instruments for continuously measuring the concen- centration must be extremely accurate and have a very small time lag.
(5) The volumetric input flow rate must be equal to the volumetric output flow rate, and both flow rates must be constant.
The theoretical results presented here have demonstrated that
different reaction rate equations do produce a distinctive set of re-
sponse curves. Therefore, assuming that the above mentioned ex-
perimental stipulations can be met, the true reaction rate equation
can be determined by this method.
30
CONCLUSIONS AND RECOMMENDATIONS
This report introduces a method for studying reaction rate
equations. It shows promise because no two different rate equations
have the same response over the complete range of parameter values.
This is especially true at lower frequencies where the higher har-
monics are more pronounced.
In order to increase the scope of this method, additional com-
plex rate equations should be proposed and theoretically analyzed;
then, experiments should be performed to evaluate the practicality
of this method.
31
BIBLIOGRAPHY
1. Black, Harold S. , Stabilized feedback amplifiers. Bell System Technical Journal 13 :1. 1934.
2. Bilous, Olegh, H. D. Block and Edgar L. Piret, Control of con- tinuous flow chemical reactors. A. I. Ch. E. Journal 3:248. 1957.
3. Eigen, M. , Methods for investigation of ionic reaction6s in aqueous solutions with half -times as short as 10 sec. Discussions of the Faraday Society 17:194. 1954.
4. Goheen, Harry E. , Professor of Mathematics. Private Com- munication. Oregon State University, Corvallis, Oregon.
5. Levenspiel, Octave. Chemical reaction engineering. New York, John Wiley and Sons, Inc. , 1962. 501p.
6. Nyquist, H. , Regeneration theory. Bell System Technical Journal 11:126. 1932.
7. Oldenburger, Rufus. Frequency response. New York, Macmillan Company, 1956. 372p.
8. Stelling, P. O. and R. B. M. Elkund, Acceleration of chemical reactions by vibration. Swedish Patent 138, 857. January 27, 1953. (Abstracted in Chemical Abstracts 47:6174h. 1953)
9. Vasil'ev, S. S. and M. S. Selvokhina, The kinetics of nitrogen oxidation in an oscillating discharge. Zhurnal Fizicheskoi Khimi1 32:1299. 1958. (Abstracted in Chemical Abstracts 53:1928 e. 1959)
10. Wight, H. M. , Influence of periodic pressure variations on chain reactions. Planetary Space Science 3:94. 1961.
Ll. Williams, F. A. , Response of a burning solid to small- ampli- tude pressure oscillations. Journal Applied Physics 33: 3153. 1962.
32
12. Wylie, Jr. , C. R. Advanced engineering mathematics. 2nd ed. New York, McGraw -Hill Book Company, Inc. , 1960. 696p.
ak
33
NOMENCLATURE
coefficient of sin (k06) 5..n general periodic steady -state solution of ri
A ratio of amplitude of inlet concentration to mean inlet concentration
Ak dimensionless output of the (k - 1)th harmonic
b mean value of general periodic steady -state solution of ri
c k
coefficient of cos (kS20) in general periodic steady -state solution of ri
CD concentration of reactant D in reactor, moles /ft3
CE concentration in reactor of another reactnt E that influences the rate expression, moles /ft
CD amplitude of inlet concentration of reactant D, moles /ft 3
fi 3
CD mean inlet concentration of reactant D, moles /ft Mi
CE amplitude of inlet concentration of another reactant, E fi that influences the rate expression, moles /ft
CE mean inlet concentration of another reactant, E, that Mí influences the rate expression, moles /ft
em' em+ l' e m + 2
error terms which include truncated and propagated er- rors at m, m + 1 and m + 2, respectively
f(n), f(r',1) dimension less isothermal kinetic rate expressions as a function of 'nand and y, respectively
gD isothermal kinetic rate expression for reactant D as a function of reactants D, E, ... , moles /ft
K, K1, K2 dimensionless kinetic rate constants
Kl
I r
K2' K4
K3' K5
34
third -order rate constant, ft 6 /moles hr 2
first -order rate constant, hr -1
second -order rite constants in mechanism of complex rate equation, ft /moles hr
Lipschitz number evaluated at m + 1 Lm + 1
N an arbitrarily large number in Eq. (18)
S stoichiometric ratio of reactant D to reactant E
t time after reactor started, hr
v., v volumetric input and output flow rates, respectively, ft 3 /hr
o
VR volume of reactor, ft3
Greek Symbols
ratio of concentration of product X in reactor to mean input concentration of reactant D
dimensionless time increment
ratio of concentration of reactant D in reactor to mean inlet concentration of reactant D
lin - 1; %' ñz + 1 in + 2
evaluated at m - 1, m, m + 1 and m + 2, respectively 1 i
in' in + 1 first derivatives of r with respect to O evaluated at m and m + 1 respectively
8 equals tv /VR, dimensionless time
(1)k phase lag of (k - 1)th harmonic
Y
D8
n
35
S2 equals u Vit/v, dimensionless frequency
frequency at which input concentration pulsed, cycles /hr
APPENDICES
APPENDIX A
DIGITAL COMPUTER PROGRAMS
(a) Numerical Calculus Scheme
1. Read necessary data cards
2. y =i - % - f(no)
3. n. = T| + ri1 A01 o o
4. e = e + AG
5. ti* = 1 + A sin £2 6 - ti - f(r) )
6- t\i =tio+ t{ri+r\\]7. Compare n to r\
(a) if(nll - ^>io"5,set ti = ti and -* Go to 5,
(b) If:(ri - r)^<10" *»Goto8.
8. Print r\ , 6
1 AO
io. e = e + AG
11. n' - 1 + A sinne - -n - f(n )
1 2. PRINT r) , 9
1 AG
36
37
14. 0 =0+ oe
15. = 1+A sin SZO -r}3 -f(13)
16. Compare 0 to 15R AO (a) If 0 = 15R A0,
calculate L0 = - 1 -afI
an 0
and R = R + 1 and Go to 17.
(b) If 0 4 15R AO - Go to 17.
17. e3 3
= (el+e2)+ 4 L0(-e1+7e2)+8(rlo- + 2-r13)
18. Print 113, 0, e3
19. 10 =711
20.
21.
22.
23.
24. el = e 2
25. e2 = e3
26. Compare 0 to MAO ( a) If 0 4 M t8 Go to 13, (b) If 8 = MAO Go to 27.
27, Compare S2 to AO
(a) If S2 4 A set S2 = 1052 and AO = 0, 1 A0, then - Go to 1.
(b) If S2 = - Go to 28.
28. STOP
1
T11 =12
38
(b) Fourier Series Solution of Eq. (5) for a Second -order Reaction
1. Read in necessary dimension statements and data.
2. j = j + 1
3, (BM2) = 1 - 2 (alk + a2.Q + a3m + a5i + c lk + c 2.2
2 ,2 + c +
c2 + c 4n 5i)
4. Compare 1 + 4K ( BM2) to 0
( a) If 1 + 4K ( BM2) 1 0 - Go to 5.
(b) If 1 + 4K ( BM2) < 0 - Go to 45.
5, bi = ZK( - 1 + 41 + 4K(BM2) )
6. Compare lb. b 1
1,to 0.00001 J
(a) If lb. bj - l
1 0.00001 Go to 7.
(b) If I bj - bj 1 < 0.00001 -; Go to 14.
7. (DM1) = SZ2 + ( 1 + 2b.K)2 J
8. (DM2) = a2.Q c 1k + a3m c
22 + á4n c3m + a5i 4n -a1k c -azec3m-a3mc4n-a4nc5i