ML20097H626
| ML20097H626 | |
| Person / Time | |
|---|---|
| Issue date: | 08/31/1984 |
| From: | Margulies T, Schwarz W NRC OFFICE OF NUCLEAR REGULATORY RESEARCH (RES) |
| To: | |
| References | |
| NUREG-0935, NUREG-935, NUDOCS 8409200389 | |
| Download: ML20097H626 (44) | |
Text
-
NUREG-0935 Acoustic Wave Propagation in Fluids With Coupled Chemical Reactions U.S. Nuclear Regulatory Commission Office of Nuclear Regulatory Research T. S. Margulies, W. H. Schwarz
- ycv9,
- y. g ';
io SsR'E RE P S4 S 0935 R PDR
NOTICE Availability of Reference Materials Cited in NRC Publications Most documents cited in NHC publications will be available from one of the following sources:
- 1. Th* NRC Public Document Room,1717 H Street, N.W.
Washington, DC 20555
- 2. The NRC/GPO Sales Program, U.S. Nuclear Regulatory Commission, Washington, DC 20555
- 3. The National Technical Information Service, Springfield, VA 22161 Although the listing that follows represents the majority of documents cited in NRC publications, ;
it is not intended to be exhaustive.
Referenced documents available for inspection and copying for a fee from the NRC Public Docu-ment Room include N RC correspondence and internal NRC memoranda: NRC Office of Inspection and Enforcement bulletins, circulars, information notices, inspection and investigation not ces; i
Licensee Event Reports; vendor reports and correspondence; Commission papers; and applicant and licensee documents and corrospondence.
The following documents in the NUREG series are available for purchase from the NRC/GPO Sales Program: formal NRC staff and contractor reports, NRC-sponsored conference proceedings, and NRC booklets and brcchures. Also available are Regulatory Guides, NRC regulations in the Code of Federal Regulations, and Nuclear Regulatory Commission Issuances.
Documents available from the National Technical Information Service include NUREG series reports and technical reports prepared by other federal agencies and reports prepared by the Atomic Energy Commission, forerunner agency to the Nuclear Regulatory Commission.
Documents available from public and special technical libraries include all open literature items, such as books, journal and periodical articles, and transactions. Federal Register notices, federal and state legislation, and congressional reports can usua!Iy be obtained from these libraries.
Documents such as theses, dissertations, foreign reports and translations,and non-NRC conference proceedings are available for purchase from the organization sponsoring the publication cited.
Single copies of NRC draf t reports are available free, to the extent of supply, upon written request to the Division of Technical Information and Document Contro!, U.S. Nuclear Regulatory Com-mission, Washington, DC 20555.
Copies of industry codes and standards used in a substantive manner in the NRC regulatory process are maintained at the NRC Library, 7920 Norfolk Avenue, Bethesda, Maryland, and are availab!e there for reference use by the public. Codes and standards are usually copyrighted and may be purchased from the originating organization or, if they are American National Standards, from the American National Standards Institute,1430 Broadway, New York, NY 10018.
GPO Printed copy price: J1.15
NUREG-0935 RH Acoustic Wave Propagation in Fluids With Coupled Chemical Reactions Manuscript Completed: August 1984 Date Published: August 1984 T. S. Margulies, W. H. Schwarz*
'The Johns Hopkins University Department of Chemical Engineering Baltimore, MD 21218 Division of Risk Analysis and Operations Office of Nuclear Regulatory Research U.S. Nuclear Regulatory Commission W:shington, D.C. 20555 1
i
n i
TABLE OF CONTENTS fage Table'of= Contents iii' List of Figures IV List'of, Tables V
cSummary 1
List of.. Symbols 1 2
I.
Introduction 5
II.
Acoustic. Theory For A Reacting Mixture 7
A.
Hydrodynamic Equations 7
8.
Stoichiometry and Chemical Kinetic Equations 8
C.
First-Order Acoustic-Eauations 12 D.
Modified Kirchhoff-Langevin Equation 13 III.
Coupled Chemical Reactions-16
~
A.
Kinetic Equations and Relaxation' Times
- 16 B.
Calculated Results 19 1.
Cobalt Polyphosphate Mixtures 20 2.
Aqueous Solutions of Glycine 21 3.
Water-Dioxane 22 IV.
Conclusion 25 V.
References 30 I
I t
I-o in
~
iii
-m w-
+
+-w y
l LIST OF FIGURES P, age Figure 1
Monochromatic Forced Plane Sound Waves of.
32 Infinitesimal Amplitude Propag '.ing Through a Fluid 2
Plots of a/f2 Versus Frequency f For Cobalt 33 Polyphosphate Solutions 3
Absorption per Wavelength Versus Frequency f 34 For 0.125 Normal Cobalt-Polyphosphate Solution 4
Plot of a/f2 Versus Frequency f For Aqueous 35' Solution of Glycine 5
. Absorption per Wavelength Versus Frequency f 36 For Aqueous Solutions of Glycine 6
Exact and Approximate Calculation of a/f2 Versus 37 Frequency f For p-Dioxane Water Solution 7
Absorption per Wavelength Versus Frequency f For 38 p-Dioxane Water Solution i
)
1 iv
f
i.i 3
s LIST ' 0F--TABLES
_P_agg
- . Table 1
Physical: Property Data For Cobalt-Polyphosphate 26'
' Mixture-(0.125N) 2 Physical Property Data'For Aqueous Glycine Solution.
27
- 3 Physical Property Data For Water-Dioxane' Mixture 28 9
I i
g i
i-t 4
V a
e-w 1
V
+
+ w w----4
=
+-ye mn-m-2 w+
e
--w a=wm-v 9
+f
i F.
I
SUMMARY
This investigation presents a hydroacoustic theory which accounts for sound absorption and dispersion in a multicomponent mixture of reacting fluids (assuming a set of first-order acoustic equations without diffusion) such that several coupled reactions can occur simultaneously.
General results are obtained in the form of a biquadratic characteristic equation (called the Kirchhoff-Langevin equation) for the complex propagation variable X = - ( a +
iw/c) in which a is the attenuation coefficient, c is the phase speed of-the progressive wave and w is the angular frequency.
Computer simulations of sound absorption spectra have been made for three different chemical systems each comprised of two-step chemical reactions using physico-chemical data available in the literature.
The chemical systems studied include:
- 1) water-dioxane, 2) aqueous solutions of glycine and 3) cobalt polyphosphate mixtures.
Explicit comparisons are made between the exact biquadratic characteristic solution and the approximate equation (sometimes referred to as a Debye equa-tion) previously applied to interpret the experimental data for the chemical reaction contribution to the absorption versus frequency.
The relative chemical reaction and classical viscothermal contributions to the sound absorption are also presented.
Several discrepancies that can arise when estimating thermodynamic data (chemical' reaction heats or volume changes) for multistep chemical reaction systems when making dilute solution or constant density assumptions are discussed.
1
LIST OF SYMBOLS
.A.
-Chemical affinity of jth reaction [ML T 2].
2 J
A).
Coefficient of change of affinity of ith reaction with degree of-9 advancement variable [M mol 1]
a Activity of ath - component [1]
y B
Frequency-dependent isothermal coefficient of expansion [1]
B Frequency-dependent isobaric coefficient of expansion [1]
e c,c Reference and frequency-dependent speeds of sound respectively [LT 1]
o c, c' Instantaneous and equilibrium heat capacities at constant pressure p
[L2 T20]
c Mass concer.cration of constituent a [ML 3]
y c, c' Instantaneous and equilibrium heat capacities at constant volume y
[L2 T.20]
C, Chemical constituent a [1]
C Frequency-dependent heat capacity [1]
C Frequency-dependent heat capacity for oth orthonormal reaction [1]
y C
Chemical constituent a [1]
y f
Frequency of sound wave [T 1]
G)
Matrix coefficients of kinetics equations [T 1]
g h
Heat of oth reaction at constant temperature and pressure [H-mol 1]
y H
Heat of oth orthonormal reaction [H mol 1]
y J
Symmetrized matrix coefficients of kinetics equations [T 1]
gj n'
Number of constituents [1]
k Wave number [L 1]
K Equilibrium constant of oth reaction [1]
y K
Bulk viscosity [ML 1 T 1]
g k,kf Forward and reverse reaction rate coefficients respectively of oth reaction [mol M 1 -T 1]
2
l k
Thermal conductivity of mixture [MLT 80 1]
e Molar density of ath constituent [mol (of a) L 3]
my M,
Molecular weight of ath component [mol (of a) M 1]
M, Molecular weight of equilibrium mixture [M mol 1]
a
-p,p,p Thermodynamic, thermostatic and acoustic pressures respectively g
[ML 1 T 23
.r Number of reactions [1]
R Number of independent reactions [1]
i R
Universal gas constant [ML2 T2 01 mol 1]
s Specific entropy [H M 1 0 13 S
Signed stoichiometric numbers [1]
gy S
Sum of signed stoichiometric numbers of oth reaction [1]
g t
Time variable [T]
v 'Vxo, v Total, static and acoustic velocities in x-direction [LT 1]
x v
Specific volume of equilibrium mixture [La M 1]
v Volumetric change of oth reaction at constant temperature and g
pressure [La mol 13 V
Volumetric change of oth orthonormal moda at constant temperature g
ana pressure [La mol 1]
Position vector and components [L]
x, x 9
X, Mole-fraction of o-constituent X
Frequency number [1]
Y Thermoviscous number [1]
Z Extent-of-react:on of oth orthonormal reaction o
Absorption coefficient [L 1]
a p,p Instantaneous and equilibrium isobaric coefficients of thermal 0
expansion [0 1]
2
p, s' Instantaneous and equilibrium isothermal coefficients of p
. compressibility [T2g.1L 1]
y, Activity coefficient of oth chemical components [1]
y=c /c Ratio of instantaneous heat capacities p y 6q Kronecker delta symbol [1]
c Specif_ic internal energy [L2T 23 th
(, (,
Degree-of-advancement vector and o component [mol M 1]
+e (y
Reaction velocity at equilibrium [mol M 1 T1 ]
no Shear viscosity [ML 1 T1 ]
a 0,0,,0 Absolute equilibrium and acoustic temperature respectively [0]
A Wavelength of acoustic wave [L]
l A
Eigenvalue of oth reaction [T 1]
g p,
Chemical potential of oth component [L2T 2,oj_13 p
Density of equilibrium mixture [ML 33 r,r*
Relaxation times of pth reaction at constant. (p, 0) and (s, p) p p
respectively [T]
X Complex propagation coefficient [L-ll w
Angular frequency [T 1]
m, w Molar concentration vector and ath component [mol M 1]
w Total molar concentration of mixture y;
Molar production vector and ath component [mol M 1 T 1]
a O
Dimensionless chemical relaxation time of oth reaction [1]
y
[
] = ' dimensions of '; M = mass (g), L = length (cm),
T = time (sec), 0 = absolute temperature ( K),
H = heat (cal), mol = g mole 4
'I.
INTRODUCTION
. Sound absorption and dispersion are directly dependent upon the properties
~
of the transmitting material and are attributed to'various dissipative mech-anisms; that is, those that contribute to'an increase of the entropy.
In particular, ultrasound measurements have provided useful estimates of kinetic and thermodynamic parameters of fast reactions'in gases and liquids which j
complement other traditional chemical kinetic measurement techniques.2_s, Combined with an historical outline of classical linearized acoustic theory, Truesdell has given an exact theoretical description for the spatial attenuation and phase velocity of_a one-dimensional, forced harmonic plane wave in a Newtonian viscous, linear heat-conducting (or Fourier), single-component fluid.10 Several investigations have extended the viscothermal sound absorption and dispersion problem for a fluid with a single chemical reaction or a binary set of reactions i
in a nonconducting fluid and have presented analytical solutions.11 14 Recently, i
the Newtonian viscothermal problem of acoustics has been solved for the case of chemically reacting fluids with an arbitrary finite number of coupled reactions l
by Margulies and Schwarz.15 Our theoretical results for a multireaction system differed somewhat from previously reported work and it is appropriate to compare the 'different methods numerically.
For example, in the analysis it was necessary to use certain transformations; that is, a reaction sequence orthonormalization to obtain an approximate formula for the absorption from the exact biquadratic 7
solution that has the simple form:
3 R
N,
,- = I
+N (1) 0 f2 4=1 1+(7 )2 t
l Previous investigators have only examined the linearized kinetic equations to obtain a nonunique coordinate transformation.
I' Experimental data are more easily analyzed by using simple functional I
formulas such as Eq. (1).
However, the range of validity of these approxi-mations, particularly the viscous term, has not been examined in any great c
detail.
This paper reports some calculations that were made in order to determine the magnitude of the various errors in these approximations and also to compare the different theoretical methods.
1 l
In this paper we present calculations of the sound absorption a and sound speed c as functions of frequency for three chemical systems.
These fluids i
are interpreted in terms of a perturbation of a two-step chemical reaction l
mechanism from a state of strong equilibrium.
5 i
.The paper is arranged as follows.
First, we discuss some background
' information and acoustic theory.
Next, the linearized partial differential equations of ~ acoustics (both ~ hydrodynamic and chemical kinetic) and the modi-fied Kirchhoff-Langevin solution are summarized.
Then, chemical kinetic equations and transformations pertinent to a relatively general two-step
- 9echanism are provided.
Finally, applications of the theory to three chemical
- reaction examples are presented.
4 1
1 6
II. ACOUSTIC THEORY FOR A REACTING FLUID MIXTURE Given a semi-infinite continuous fluid medium initially at rest, the y, z plane of. fluid is oscillated harmonically in the x-direction with frequency w as shown pictorially in Fig. 1.
This displacement of the plane boundary and resulting longitudinal wave motion within the fluid is assumed to be infini-tesimal.
The wave amplitude progressively diminishes with transmission j
distance as the mechanical energy which is imparted by the boundary surface is dynamically converted into thermal energy.
This analysis assumes that the actual motion should be sufficiently well approximated by the exact solutions to the first order field equations of the mixture, in which all nonlinear terms are omitted, and moreover, all coefficients are evaluated at their uniform equilibrium reference values.
The total value of each variable, I, in the list of system variables chosen to define this one-dimensional perturbed motion is decomposed into an equilibrium (static) and time-dependent incremental acoustic contribution.
I (Variable),7o (Variable) ya (Acoustic)
Total Static Variable a
a a
each acoustic variable, I, such as pressure p, temperature O, or Furthermorg,is represented as a damped sinusoidal progressive wave by velocity v a
la = Re {I exp (xx + iwt)}
(2)
X = - (a + i k) is the complex propagation variable where a is the spatial absorption coeiiicient and k is the magnitude of the wave number vector perpendicular to surfaces of constant phase.
k = w/c is the spatial analog of frequency and represents the number of wavecrests per unit length, or simply 2r1 multiplied by the reciprocal of wavelength in this one-dimensional problem.
A.
Hydrodynamic Equations For waves propagating in the x-direction into a fluid mixture initially at rest with a uniform equilibrium state the hydrodynamic equations for a Newtonian viscous fluid with Fourier heat conduction may be written in terms of the acoustic variables as follows:
DO BQ Bf av
~EO 5t O
'E
- j at3 l
a p at O
a a
2 a g g + ap ~ (4
+K)3y
.= 0 (4) av G
p ax 3 o g p l
"7
[
r h)
-k
=0 (5) e, se o
i P C P
e o p j = 1, 2,.. R reactions.
These equations represent the linearized balances of total mixture mass, linear momentum and energy when diffusion and radiation supply can be omitted.
Definitions for these variables and their units may be found in the nomen-clature section.
Repeated indices should be summed except when they are printed as bold subscripts (or underlined).
B.
Stoichiometry and Chemical Kinetic Equations Here, we consider a general reaction scheme composed of an arbitrary finite number of elementary reaction steps, say r, involving a = 1, 2,... n constitu-uents C.
This roay be expressed by y
n I
S C,
=0 p = 1, 2,.. r (6) a=1 py where the positive values of the stoichiometric coefficients S correspond to pa the reaction products while negative values correspond to reactants.
Also, by convention, a neutral species such as an inert solvent is assigned a stoichio-metric number of zero.
Let the rar.k of the stoichiometric matrix be denoted by R 1 r.
This determines the possible number of independent reactions of the system.
As a consequence, the total collection of reaction sequences may be completely represented by a selected subset of R reactions.
It is noted that the number of " independent reactions" determines the calculated number of relaxation times.ts In order to express the composition changes during the progress of a reaction in a nondiffusing mixture, it is convenient to define the degree of advancement such that e=g3 a = 1, 2,..
n (7)
,_m a
a p pa p = 1, 2,... R where m' is a reference molar density, say at equilibrium.16 A concise mathe-matical treatment of the stoichiometry of reacting materials has been given and provides a natural means for introducing the reaction velocities 17 and gives a proof of the Law of Definite Proportions.18 This is briefly discussed below starting with an assertion that the atomic substances making up the mixture of constituents are indestructable which may be expressed by the equation n
q g
I T,,
_g
=0 o' = 1, 2,.. U elements (8) q=1 g
i 4
8
9 T,' represents the number.of moles of atomic substance a' in one mole of the E
q th constituent.'
Also, note that-if we multiply by W",,'the atomic weight of a', and sum then Eq. (8) obtains n
+
U
=I T9 I
c
=0 since M
a'=1 ", W",
9 q=1 q
Moreover, Eq. (8) establishes.the following relationships among the n mass supplies
+
n-K
(*){+
pH I
S c
=
9&
9 9 4=1
+
n-x N
4 f
9$
41 where the stoichiometric matrix [Sq ] is any nx(n-x) matrix of rank (n-k) E R such that n
l' 2'
- (" k)
I T 9 S
=0;
","_ 1, 2,.. U y
q=1 a'
q$
and x = rank [T 9] 5, minimum (n, U).
is interpreted as a reaction velocity y
or rate.
If Eq. (10) is multiplied by W, and summed over o' y
U n
n 9
I I
T, W,
S@
=
I M S
=0 (11) 9 9 94 a'=1 q=1 q=1
'This latter expression corresponds to the notion that the (n-K) reaction equations are balanced.
Neither the stoichiometric matrix nor the reaction velocities are unique; however, the maximum number of linearly independent chemical reactions possible in the mixture is unique.
In our acoustic and l
appliej chemical kinetic analysis the reactions are assumed in advance.
i Therefore this defines the elements of the stoichiometric matrix to within a nonzero constant.
The equations of mass balance for species a (i.e., sometimes called the partial mass balance equations for the mixture) may be written as 9
r i.
+-
R-
.+
S, (,
q = 1, 2,.. n m
=
w
='
q
. q q
.or alternatively
+
(4
.(,
$'= 1, 2,.. R (12)
=
which gives the composition of the mixture. constituents versus time.
Assuming a polynomial mass-action form for the kinetics constitutive relation the chemical kinetic (or partial mass balance) equations _ become:
S*
+
n S.
n
()
.()
k (0,p) n a
k (0,p) n a 'd (13) 0J
=
=
y y
a=1 a=1 j = 1, 2,.. - R S}. ands}denotethepositivestoichiometricconstantsforthereactantsand y
y products,.respectively.
Further, a linearized version of the above equation can be derived for small departures from a state of strong equilibrium.
i i
T j
a(X /R e)*
a(X /R e)'
a(K /R O)*
3 3
j
{*E 80 3~0a
+[
3 a+[
ap p
gC,
]C (14) a I
j l
I-where
+
n
('
=k D -(a')baj = k H (a')b+j n
a a=1 a=1 l
yj_
yj _ S+j)
(S S
=
y i
i a(X /R e)'
h.
a(X /R e)*
~ v.
3 3
5 3
[
]= R 03 ' [
]=
30 T
ap i
Rs o T
a(X /R O)*
(Sjy gy)
S j
3 l
-[
]EAjg =.M, [ I
-S S ] + NI j g 0(g a=1 x I
a i
1 10 l
~
- i
_1 3
-(Anfy)-
'NI's M, -Sjy y
the activity coefficient of a).
ag =1y X is' the activity.of component a (with yg
.y u
- For' ideal. systems.y = 1 and the non-ideal-terms, N.I., are identically zero.'
y f
Equation (14).is a system of coupled linear differential eq'uations of first' order.in'(*J.
For e =
a= 0 and for a' single reaction-(i.e., j=1), a-a chemical relaxation time towards equilibrium can be defined as (aX e-
[
'l iy
=
i i
LTt Re -DC o'
1
- a 1
a g
such that-
.(.3 T1
-I Wh'en considering multistep' reaction systems each chemical relaxation time rather than corresponding to an isolated reaction step corresponds to'a
" normal mode" of the chemical system.
Also,' for a system at constant {p, 0}
the kinetics equations become-
~
jg (a
=-G
+
l where Gjg = (j'A and the matrix Q is not necessarily symmetric.
jg A coordinate transformation of the linearized kinetic equations (at constant 0 and p) is introduced at this point in the analysis in order to separate the differential equations-into disjoint, independent equations.
Let
+
-(a -= (g )4 a g
J l
Then Eq. (14) becomes L
.(j=-Jjg-(a (15)
+
+
((*)b = J p=((})b
.w ere J
.A h
jg gj 11
M Now it is convenient to define:
(* = M Z
- . j, o = 1,-2,.._R yy y
such that 1
I (M
J f
gj jk "ka)
(16)
= -
o D
Z
=-A Z
.=
g gy y y
g and M 1 y d = Q is diagonal.
Its elements are formed from the eigenvalues
{A } o = 1, 2,.. R obtained from det (A S
- ok) = 0.
y ok The reciprocals of the elements of the d{ agonal matrix represents the relaxation times of the reaction (i.e., t
= p ).
It can be shown that:
y o
(1) both the eigenvalues of the matrix Q and J are the same; (2) the matrix y needs to be orthogonal (i.e., M 1 = M ) as previously shown T
to obtain sound absorption formulas similar to Eq. (1) for practical application.1s (3) different relaxation times may be defined according to the particular set of independent variables used.
For example, equivalent relaxation times can be defined with {0, s} held fixed where s is the specific entropy.
C.
First-Order Acoustic Equati.ons The linearized Eqs. (3, 4, 5 and 14) explicitly written for a two independent reaction mechanism are I+V 2] + '
=0 (17)-
+ P,g po [V i t
2 p
a a
8y2a av
+ gp
- (K +3q)7=0 (18) 4 po at ax g
9 p( ap*
- Po [H Z
+H Z]-kep=0 (19) as a2e' P C
-O o p,[ at g
e, at 1
2 2
~
12
a V,
a i + "'
Zi=0 (20) e p '+ At 3
i T
Re$~
Reo
+"{08 O
pa+A Z2=0 (21) a 2
2 R
R Oo These equations constitute a set of fir:,t-order hydroacoustic equations for infinitesimal sound waves propagating in the x-direction.
These equations admit a
a a
damped harmonic waves of the form of Eq. (2) where la={O,
,y
, Z, Z }-
3 2
For nontrivial X solutions, the determinant of the matrix of the coefficients, given in partitioned form below,.must be identically zero.
~\\
Ia B
(8-R41 G (22) det
~
=
0
.=
det
\\c a>
where 8=
-iwP I*0 X
O p
0 x
ipw - (K +fq,)X2 g
(impc - k x ) -iwe p 0
p e
ge
\\
{l+iwI g=
[-imp,V 0
-imp V2 t
i g
0 0
a=
(
0 1 + iwT imp H iwp H 2
3 2
g g
f H'
h o
g=
T T
R ef Reo H2_.
Y2 0) i T
Re$
Re o D.
Modified Kirchhoff-Langevin Equation In general, a complex-valued biquadratic algebraic equation for X is obtained.
This may be expressed in terms of dimensionless parameters as follows:
13
(
)4 [iXY (iyXB*+1)] + (
)2 [ +iyX (C" B" Y-1 (g )2 + y g )]
w W
+ y [C* B*
y-1 (B")2] = 0 (23)
P P Y
P where k = w/c, g
4 X
= w(3 q, + K )/p c8 Frequency Number g
4 K)c Thermoviscous Number Y
=ko (3 9
+
0 g
p O
=Tm Reaction Frequency Number g
g and y
= c /c Ratio of Specific Heats p y
- Also, 2
R AC "g
Io pg Cm=1+
1 AC
=
po i
P g=1 1 +j,7 c Re2 p
g H V T R As
- y g-g -
B" = 1 + 1 AB
=
o=1 1 + iwto v
0 o
r I
R AB o
B* = 1 + I 3g o-PG
=
P i
P g=1 1 + j,7 vp R eg
+
+
((0* )b V
= ((2)
EP M
o M
h V
H =
EP O
P P
For the cases of 1) a single-component nonreacting fluid and 2) a fluid mixture with a single elementary reaction the algebraic equations correspond to the results of Truesdellto and Mazo11, respectively.
In the case of a nondiffusive, 14
4 s
1,'C
+ 1, By + 1) the' classical Kirchhoff-Langevin nonreact ng: system (B i
4 equation-is obtained.
-(f-)4-[iXY(1+iyX)+(f-)2 (1 + iX(1+yY)] + 1 = 0 o
o The propagation constant X can be determined by solving the complex algebraic equation numerically.
Two of the four values of X are unphysical (negative values of a) and the other two' correspond to type I and type II waves respectively.
The type I waves are chosen to correspond to the classical theories.
The type II waves are found to have very large attenuation and have not been observed experimentally.
15 w
m-
--y Fr
III.. COUPLED CHEMICAL REACTIONS A.
Kinetic Equations and Relaxation Times The ideas and formulas in the previous section will be specialized to the case of two independent coupled chemical reactions and later c:ad in the illustra-
.tive acoustic perturbation examples.
Consider the two-step reaction mechanism S$3.C +S$2C sac 3 2
L S
C
+S C + S a Ca + 5+2, C, + S+s C 21 3
22 2 2
2 5
~
where C represents chemical species a = 1,5 and S and S denote the posi-y py y
[
tive stoichiometric constants for the reactants and products, respectively.
I The stoichiometric matrix S for this relatively general two-step mechanism pa becomes:
[-S$1
-S$2 Sa 0
0 S
=
l (24)
S s)
~
~
-S$3
-S
-S S
22 23 24 The Rank (Spy) in Eq. (24) equals two (i.e., R=2) corresponding to two independent reactions and 5
= I S
obtains 5
P a=1 P"
(-S$1
-S$2
+ S a)
S
=
i S$2
-S$3
+S
+
5
(-S$1 4
S s) 2 The linearized kinetic equations (at Oa=pa = 0) for this two independent reaction example are written Ar (i At2 \\
($
f (7) f(
i (25)
=
+
+
( ($ j
((
A
(
A
((
21 22 Using the mass-action kinetics constitutive relation and assuming an ideal mixture (e.g., y, = 1 ; a = 1, 5) further defines the kinetics problem:
16
+
+
k((0,p)(X)b (X ) 12 kf(0,p)(X)13 (i
=
=
3 i
2
+
R (0, p)(X )b+d X )b+
($=k[(0,p)(X)bM X )Sn(X )S23 = k 3
2 3
2 4
3 n
[ (b ')
(
2)2. (3- )2 _ (
)2]
j A
+
=
tt
[(b ')(b;') + (SI2)(S$2)
(S 2)(S$2)_(3)(3)3 A
=A
=
12 21 X
X X
1 2
i 2
3
~21)2.(g;2)2.(s;a)*+(S 4)2.(gl3)2 (S
=[X X
X X
X
- (5 )23 A22 2
i 2
3 4
3 Also, Eq. (15) gives the transformed reaction velocities
\\
IJ J
\\
I (i\\
., i ii 12
=
(26)
{ (2, (J 21 22 j
( (21 J
where an orthogonal matrix d is used to diagonalize }.
s
(*A J
=
ii tt
+
+
(Ci ($)'1 J
J A
=
=
12 21 12
+
d (2 A 22 22 17
The relaxation times are computed from B
\\
[ Jgi -A J t2 0
det
=
J J
-A 21 22 A
A2=
{ tr y
[(trd)2 _ 4 det y]b)
(27) or
= f tr y {l
[1-4 (det J/(tr y)2)]}
+A and det J = At A-
- A2=
tr y = At where ) =
2 2
The chemical system relaxation times, in general, are not simply related to the individual relaxation times, say Igg and T if it were possible to 22, set up the equilibria in each reaction (steps one and two) in isolation.
Let
+
-($A s
tt Igg
+
1
- (* A s
22 T22 4 det J For
<<1 or when the relaxation times are well separated
)2 b
1--
~
It
- tr J-Igg
[22
+
=
det ]
1 T2 tr y and the first overall relaxation time is equal to the single very short time.
18
- Now, I (a )
Mai M
IZ t2 1
{ l$ j kM 2
22 j (Z )
H 2
The components of M for a two reaction case are obtained from M
J,,,
A, - J,2 2
=
=
M A - Jgg J
21 t
21 "12 d '>
A
-d i
9 2
and M-
=
=
A
-J J
2 11 21 22 2
2 2
2 1)
+M
= 1 and M12 + M22 When conditions of orthonormality (e.g., Mit
=
21 are used, the orthogonal transformation matrix becomes 32,
]g
=
M**
J2, g
(A2 - Jai)2 y=
[(A J52 Jit) + Ji2
+
t J,,)2 g
E((A, - J,i)2
\\
Ag
-Jit)"
~
+ (A2 A - Jtt)Z + Ji2 J{2 t
B.
Calculated Results Estimates of the absorption or sound speed versus frequency may be directly calculated given that all the physicochemical parameters that define the materials under study are known.
Alternatively, measured absorption data, for exar. le, when combined with data available in the literature for some of the parameters can be used to quantify unknown parameters associatd with fast reactions (such as kinetic and thermodynamic information).
To this end, simple formulas and direct relationships such as given by Eq. (1) are sought for fitting the data from experimental measurements.
N, (a>o) in Eq. (1) is proportional to Q2 whic' is a linear combination of the heat of reaction and the volumetric change for the orthonormal reaction sequence.
Our relations between the normalized reactions and the proposed two-step reaction sequences are given as:
+
+
(($)b
(($) M21 42 (28)
M Qi it 4 +
=
19
+
+
Q2
= (( ) M
(( ) M 21 91 +
22 42
- o = 1,2 where q
=h
(
)v g
y g
Further, our approach obtains N, = 2fla I, (Y pc 0
H)
(29)
~
~
a c
a T
~
P -
o It is noted that this relation differs from other investigators because of the transformations required in the acoustics equations to obtain the simple approximate forms given by Eq. (1).
For the single reaction case the comparison is exact.
We remark that the theoretical derivation used by others did not explicitly use the balance of mass, linear momentum and energy for acoustical wave propagation and was obtained in a heuristic manner.
We wrote a computer program to calculate the absorption and speed of sound by the " direct" method, which requires all the physical property data.
The results of calculations for three chemical systems including cobalt polyphos-phate mixtures, aqueous solutions of glycine, and water / dioxane are discussed in the following sections.
1.
Cobalt Polyphosphate Mixtures The technique of ultrasonic absorption has been used to study counterion site binding for polysalt solutions such as cobalt polyphosphate.
Arguments were made that 1) the absorption mechanism is analogous to Eigen's ion pair formation in simple electrolyte solutionsa and 2) a fraction of the ions were condensed according to Manning's theory for polyelectrolytes.19 The model considers ion pairs exhibiting three states of hydration that were in equilibrium relative to the two reaction sequence:
A B
C (outer sphere)
(inner sphere)
State A corresponds to the hydration shells of the polyion site and the counter-ion in contact without overlapping; State B consists of relatively unaffected counterions and partially dehydrated polyion sites and State C consists of the hydration shells of both the counterions and the polyion sites being modified.
Zana and Tondre20 2: have examined the divalent cobalt polyphosphate system in some detail using density, nuclear magnetic resonance and ultrasonic measure-ments.
They assumed 1) that the ultrasonic absorption due to chemical reactions is only affected by the condensed counter-ions and not by the mobile ions; 2) the ultrasonic absorption is determined primarily by volumetric changes and not by 20
,.. _. ~
the heats of reaction; 3) the fraction (f ) of condensed counter-ions can be calculated from the work of Manning
= M*A * "B + "C f, MA and is independent of the polyion concentration provided it is not too large;
- 4) the excess absorption due to the above describad chemical processgs can be separated from other effects by subtracting the obsorption using TMA (tetramethylammonium) as the counterion; and 5) the forward rate for the second reaction (B ; C) which relates to the exchange of water molecules from the inner hydration sheath of the ion and bulk water is close to that of any small ligand and is given by 1.333 x 105 mol g 1 sec 1 In our calculations, we have made certain assumptions:
- 1) the degree of polymerization is not important provided that it is large enough (P >500);
- 2) the equivalent monomer molecular weight.is 108.43; and the solutions are sufficiently dilute such that the values for the physical properties are close to pure solvent (water).
Refer to Table 1 for a list of physical property data used in the calculations.
We remgrk that the approximate calculation of a (cobalt polyphosphate)/f2 minus a(TMA -PP)/f2 differs from the exact by less than 1 percent which is well within experimental error.
The relaxation frequencies did not change over the concentration range (0.068 N to 0.125 N) examined.
(Refer to Fig. 2).
The relative contributions to the absorption per wavelength (aA curve) are shown in Fig. 3.
The viscous contribution is less than the chemical reaction contribution at low frequencies but dominates at higher frequencies.
The first reaction is fast and corresponds to the second peak while the second reaction is slower (first peak).
The relaxation frequencies are about a factor of ten apart.
Approximately 1/5th of the second reaction peak is contributed by the first reaction.
Total [(aA)1 + (aA)2] maxima may not correspond directly to the relaxation frequencies even in this well separated case.
The reaction volume changes are estimated to be approximately equal to the normal reaction changes.
Zana and Tondre reported reaction volume changes of approximately a mol 1 for reactions one and two, respectively.
Our data-fit 22.9 and 4.4 cm and interpretation (using the orthonormalized approach) gives 20.98 and 5.26 3 mol 1 for the volume changes associated with the first and second reaction.
cm 2.
Aqueous Solutions of Glycine Zinov'ev et al.22 have reported their measurements of aqueous solutions of glycine (0.53 moles liter 1; pH 10-13) for temperatures 15 to 30 C, in the frequency range 0.5-1200 MHz.
The data were analyzed on the basis of a proton transfer reaction of the form:
0
+
E C
A
+
B 4
+*
where A = NH CH 000'_ ; B = OH' ; C = intermediate product, 3
2 D = NH CH C00 and E = H 0.
2 2
2 21
t 4
This proposed mechanism is' consistent with the two relaxation shape of the absorption spectrum.
Previous investigations have identified a single acoustic
. relaxation ~(dependent on pH) and Appelgate postulated a two-step reaction to explain the estimated singla reaction volume change.2s.2s Data were fit to
-the approximate formula givu by Eq. (1).
We have applied our calculation procedure to Zinov'ev et al.'s data for,25 C using values for physical properties listed in Table). 'See Figs. 4 and 5.
The rate constants reported were used but adjusted for M, and a'slightly different value for the' classical absorption was used.
Our calculated results for the reaction volume changes are 25.07 and -0.444 cm mol 1 for reactions one and two, respectively.
The 3
first reaction volume change corresponds almost exactly to that of Zinov'ev et al. (e.g., is within experimental measurement); however, the second reaction volume estimate is about a factor of five different (in comparison to their estimate of -2 cm3 mol 1).
This descrepancy in the second. reaction volume change which results from the orthonormal. transformation occurs even for this case when the relaxation times are well-separated.
3.
e tar-Dioxane Hammes and Knoche2s reported their acoustic experiments for the water p-dioxane system and interpreted their' absorption data by postulating the coupled independent two-reaction system 2W
+
D DW (Q) 2 D
+
DW2 DW (Z) 2 2 The authors used a " trial-and-error" method to calculate the coefficients in Eq. (1) with the relation 2
2 f
N, = 2f1 pcoI rQ
/R 0
[VM,(4/X*+1/X[+1/X'-4)
],
where F =
t IVMe [1/X' + 1/X* + (ai2/aii)/X' + 1/X* - (a /aii)/k' r2 2
(2a 2n /a31 + 1)]}
and Q=V - (p /pc ) H e
p There appears to'be a basic difference in the theoretical results.
It is noted that Hammes and Knoche only examined the linearized kinetics equations to obtain a nonunique normal-coordinate tran? Formation.
This is not the same transformation necded to obtain Eq. '(1) fru. the formal theory.
Hammes and Knochers obtained n
Hi=- h (30) i H2=- (a12/ats) h
-h
. 3 2
22 m
and similarly fw the. volumetric change.
Here the H 's and h 's _gorrespond to g
p
>the heats of reaction for the normalized reactions and the proposed reaction sequences, respectively.
' agi = ~ - k( (X')2 - 4 k( X' x' - k( + 4 k( (X')2 (X{}
and '-
ata=k[(Xh)2 (1-2 X') - k(
where the k's correspond to the reported kinetics cons} ants of Hammes and Knoche.
We also remark that our kinetic constitutive relation (p (the molar production term for the pth reaction) is in terms of the degree-of-advancement variables w' = [ Spy (p (w is the molar concentration-moles of a/g of (p where my a
solution).
The consequence is that the reaction rate constants used by Hammes and Knoche differ from ours by the factor Me (the molecular weight of the mixture at equilibrium).which is concentration dependent.
F The parameters for the exact calculation were obtained by:
- 1) choosing kg (ork()andk[(orkh)andusingthevaluesofHammesandKnochefortheequi'i-brium constants (K and K ) in order to obtain eigenvalues of the two reactions 1
2 that matched the experimental data; 2) choosing values for the qj (or linear combination of heat and volume change) for the actual reactions to fit the absorption-frequency data so that the calculated values of N. matched the J
measured values; and 3) selecting a value for the bulk viscosity of the mixture to match the experimental data at high frequencies where the viscous contribution to the absorption dominates.
The other physical property data are listed in Table 3 for Oo = 25 C and X,
= 0.62.
Using these parameters, the absorption was computed with the exact formulas and is shown in Fig. 6 compared to the results of the approximate formula.
The values are only about i
2% higher, and within the error of experiments.
The contributions for each reaction and the viscous and thermal effects are shown in order to indicate l
their relative magnitude (Fig. 7).
It is obvious that the viscous term is not small compared to the reactive terms.over the entire range of frequencies.
We i
remark that the approximate formula for the viscous term is valid at frequencies such that wt >>1.25 From these calculations it is not possible to determine p
if the difference between the exact and the approximate values is due to the viscous approximation or the chemical approximation.
Because the criteria for
'the chemical approximation are satisfied and the error decreases at high frequencies where the viscous term dominates, we attribute the difference to the nonlinear viscous contribution at low frequencies.
23
Assuming values for the normal heats of reaction reported by Hammes and Knoche (i.e., 11000 and 1300 cal mol 1 for reactions one and two, respectively) i Twhich neglects any volume change contribution, the_ individual reaction heats would be calculated according to them by Eq. (30).
Their results give approxi-mately the same value for the heat for the first reaction while the estimate for the second reaction heat differs somewhat.
This descrepancy is partly due to the normalization of the transformation J used to separate the chemical kinetic equations.
Our calculated results for the individual reaction heats are 11010 and +215 cal mol 1 for the first and second reaction, respectively.
It is noted that the signs of the heat (or volume) changes for two independent reactions occur as (+,
-) or (, +) pairs corresponding to our choice of det M = 1.
-The pairs.(+, +) or (
, -) would correspond to the transformation having-the property that its determinant equals negative one.
Also, it is noted that our reaction rate constants are slightly different than Hammes and Knoche, even though the equilibrium constants and eigenvalues are the same.
1 l
[
24
CONCLUSION The absorption and speed of sound have been calculated for three different chemical systems by the " direct" method which requires all the physical property data, in addition to, the approximate formula for the absorption - sometimes referred to as a Debye equation.
The assumptions needed to obtain the approxi-mate formula are15 1) viscous, thermal and diffusive effects are negligable
- 2) either a dilute solution or S
=0
- 3) a < < k and an ideal solution.
o
-i Furthermore, the transformation must be orthonormal (i.e., 8 M where det M = + 1.
-- =1;M1 =g)
The exact calculation and interpretation of the sound absorption spectra for the chemical systems examined can provide different estimates of thermo-dynamic reaction parameters compared to the approximate formula, especially for estimates of thermodynamic parameters associated with the second reaction.
In general, the calculated results depend explicitly upon the orthonormal transformation M.
Our examples indicate that the approximate formula gives estimates of thermodynamic parameters for the first reaction which closely agree with those derived from the exact calculation.
However, the calculated second reaction volume (or heat) change appears to sensitive to the orthonormal transformation used even in dilute cases where the chemical relaxation times are well separated.
Furthermore, assuming that the viscous term can be decoupled from the kinetics by a linear subtraction is not necessarily valid as shown in Fig. 7 and previously discussed.
The practical consequence of a large viscous contri-bution is the loss of sensitivity of the acoustic data for determining the kinetics contribution.
All in all, ultrasound measurements provide extremely valuarle information on the thermodynamics and chemical kinetics of fast reactians in solutions.
Approximate formulas for predicting and attributing sound absorption and dispersion due to chemical reactions must be shown to be quantitatively valid.
l l
l I
l 25
4 Table 1:
Physical Property Data For Cobalt-Polyphosphate Mixture-(0.125N)
' Calculation r
.f,(bound fraction) = 0.86' Oo.
= 298.16 K k(
=.273 x 107 mol g 1 sec 1 kh
=.00455 x 10 7 mol g 1 sec 1 p,- ' = 0'.997'g cm 3
.kh
=.00950 x 107 molfg 1 sec 1 kh
=.01583 x 107 mol g 1 sec 1 S
= 0.000257 K1 e
K
=. 60.
i c
= 0.99828 cal g 1 oK 1 K
= 0.6 p
2 c
= 0.9878 cal g 1 OK 1 y
q
= 0.0100 P T1 1.982 x 10 8 sec
=
o
_7 k
= 0.0266 P T2 2.188 x 10 sec.
=
g
~
c,
= 1.497 x 105 cm sec v3
. = 20.98 cm3 mol 1 3 mol 1
= 5.26 cm v2 Ng
= 107.0 x 10 17 cm.t sec2 1445 x 10 17 cm 1 sec2 N
=
2 24.7 x 10 17 cm 1 sec 2 No
=
1 4
1 6
)
26 i
w Table,2: :PhhsicalProperty~DataForAqueousGlycineSolution I
.L p,, '= 11.6 m
= 5.2 mol 2 1' H
.0o
= 298.16 K
'm
= 5.2 mol 2 1 5
.997 g cm.a K
= 1.7 x 10
=
1 pg
~
E
=. 000257 K1 K
= 10.4 2
0 c
=.99828 cal g 1 OK 1 k(
=.472 x 1012 mol g 1 sec 1 k[
= 1.-7 x 1010 A mol 1 sec 1 OK1 kf
=.5 x 108 mol g 1 sec 1
.98784 cal gm 1 c
=
y if
= 10. x 106 sec 1
~k
= 1.4 x 10 3 cal cm sec 1 K1 kh
=.317 x 108 mol g 1 sec 1 e
ih=5.7x108 sec 1 kh
=.333 x 105 mol g 1 sec 1
.008904 P
=
q, ih=0.6x106 sec 1
.0237 P v1
= 25.07 cm3 mol 1 K,
=
18.61 g mol 1 v2
=.444 cm3 mol 2 M,
=
3 = 5.7 x 10 9
= 1523 m sec 1 1
sec c,
(H O) = 1496.7 m sec 1 c
2 g
2 = 1.5 x 10 7 2
Ng
= 100 x 10 17 cm 1 sec T
sec 2
N
= 270 x 10 17 cm 1 sec 2
No 2
23 x 10 17 cm 1 sec
~=
glycine
[
M "C
"D E
A B
79.11.
17.
96.11 79.11 18.00 80.11 o
27 E
r t
Table 3:
Physical Property. Data' For Water-Dioxane Mixture Quantity /Value Source M
= 0.62 (25% water; 75% dioxane)
Oo
= 298.16 K c,_
'= 1.48'x 105 cm sec 1 Atkinson et al. (Table III)27 p
= 1.0361 g cm 8 Atkinson et al. (Table III)27 g
c
= 2.6 joules g 1 OK 1 Stellard and Amis 2a p
=.621 ergs g 1 OK t
. y.
= 1.18671 Calculation:
y=1+0opjc/c 2
p k
= 8.628 x 10 4 OK 1 cal cm sec 1 Atkinson, et al. (Table III) e q,
= 0.02 P Emel'yanov et al.29 (adjusted fo-temperature)
K
= 0.0503 P Chosen to obt in (a/f2)
N g
g kg
= 150 x 10s mols g 1 sec 1-k(
= 140 x 107 sec 1 N
30 x 105 mollg1 sec 1 k
=
i
-R k l 28 x 107 sec 1 Hammes and Knoche2s
=
F k
= 24 x 105 mol g 1 sec 1 2
i kh
= 15 x 107 sec 1 l
16 x 105 mol g.1 sec 1 k
=
2 L
k
=
R 10 x 107 sec 1 2
l Ag
= 7 x 108 sec 1 Hammes and Knoche2s 1.9 x 108 sec 1 A
=
2 l
28 G
.w
=vne
Table 3.
(Continued)
K
=5 Hammes and Knoche2s t
K
= 1.5 2
N
= 30 x 10 17 cm 1 sec:
Hammes and Knoche2s 3
N
= 12 x 10 17 cm 1 sec2 2
No
= 47 x 10 17 cm 1 sec2
- Atkinson et al. lists 6.338 joules g 1 OK 1 for X = 0.5 which differs from MorcomandSmith'slistedvaluefortheexcesshedtcapacityandStallardand Amis' value of 2.25 joule g 1 OK 1 at 40 C.
29
REFERENCES 1.
K.-Herzfeld and T. Litovitz, Absorption and Dispersion of Ultrasonic
' Waves, (Academic Press, New York, 1959).
2.
R._T. Beyer and S._V. Letcher, Physical Ultrasonics, (Academic Press, New York, 1969).
3.
M. Eigen and L. de Maeyer, " Relaxation Methods", in Technique of Organic Chemistry, 2nd ed., edited by S..L. Friess, E. S.-Lewis, and A. Weissberger, (Interscience, New York, 1963), Vol. VIII/2, pp. 895-1054.
4.
C. F. Bernasconi, Relaxation Kinetics, (Academic Press, New York, 1976).
5.
V. V. Markham, R. T. Beyer, and R. B. Lindsay, Rev. Mod. Physics M, 4 (1951).
6.
D. Sette, Encyclopedia of Physics, XI/1 Acoustics, edited by I. S. Flugge (Springer-Verlag, Berlin, 1961).
l 7.
H. J. Bauer, J. Lamb, J. Stuehr and E. Yeager in Physical Acoustics, Vol. IIA, edited by W. P. Mason (Academic Press, New York, 1965).
8.
A. D. Pierce, Acoustics:
An Introduction to Its Physical Principles and Applications (McGraw-Hill, 1982).
9.
J. Steuhr, " Ultrasonic Methods," in G. G. Hammes (ed. ), Investigation of Rates and Mechanisms of Reactions, Part II (Techniques of Chemistry, Vol. 6, 3rd ed.), Wiley, New York (1974).
10.
C. Truesdell, J. Rat. Mech. Analysis 2, 643 (1953).
)
11.
R. M. Mazo, J. Chem. Phys. 28, 1225 (1958).
12.
L. S. Garcia-Colin and S. M. T. DeLa Selva, Physica 75, 37 (1974).
13.
J. Meixner, Scustica 2, 101 (1952).
i 14.
J. W. Nunziato and E. K. Walsh, J. Acoust. Soc. Am. 57 (1), (1975).
15.
T. S. Margulies and W. H. Schwarz, J. Chem. Phys., 77, (2), 1005, July 1982.
16.
T. E. T. deDonder, in Lecons de Thermodynamique et de Chemie, edited by F. H. Van den Dungen and G. Van Lergerghe, (Gauthier-Villars, Paris, 1920).
17.
R. M. Pov n, Arch. Ratl. Mech. Anal. 24, 370 (1967); J. Chem. Phys.,
49, 1606 (1968).
18.
I. Prigogine and R. Defay, Chemical Thermodynamics transl. by D. H.
Everett) Longmons Green and Co. Inc., New York (1954).
30
19.
G. S. Manning, Q. Rev. Biophys. 11, 179, 1978; J. Chem. Phys., 51, 924, 933 and 3249, 1969; Biopolymers, 9, 1543, 1970.
20.
R. Zana and C. Tondre, Biophysical Chemistry, 1, 367, 1974.
21.
C. Tondre and R. Zana, Polyelectrolytes, 323, 1974.
22.
O. I. Zinov'ev, Yr. M. Kozleuko, K. V. Kuranov, L. F. Pugatova, and M. S.
Tunin, Sov. Phys. Acoust., 28 (4), 299, July-August 1982.
23.
K. Applegate, L. J. Slutsky and R. C. Parker, J. Am. Chem. Soc. 90, 6909 (1968).
24.
M. Hussey and P. D. Edmonds, J. Acoust. Soc. Am., 49, 1309 (1971).
25.
G. G. Hammes and C. Nick Pace, J. Phys. Chem., 72, 6,~ June 1968.
26.
G. G. Hammes and W. Knoche, J. Chem. Phys. 45, 11, p4041, December 1966.
27.
G. Atkinson, S. Rajagopolan and B. L. Atkinson, J. Chem. Phys., 72, 3511, 1980.
28.
R. D. Stallard and E. S. Amis, J. Amer. Chem. Soc., 74, 1781, 1952.
29.
M. I. Emel'yanov, I. N. Nikolaev and F. M. Samigullin, Zhurnal Struckturnoi Khimii, 12, 161, 1971.
d 31
l t
A f0h
{f8 ygh Yn a
- X l = lo exp [-(o + h)x + iwt]
c, X
X=
Cornplex Propagation Constant o=
Attenuation coefficient w=
Angular Frequency (= 2 nf) c=
Sound Speed i
Figure 1: Monochromatic Forced Plane Sound Waves of Infinitesimal Amplitude Propagating Through A Fluid 32
eo
~
1 I
a Co
'5o "o
m W
95 e
o o g&
m x
m a
9 x
,m m
o
\\\\
~Eb E
E e m
~
a zo
- o u s
z
> o o
u.
m O
z.
n W
o D x E.
0w
[
_?
u c-*
E u.
m U
O O
o o t x
b N
N 2
s n
O ll E
m
~
9o z
5 e
o L
3 Q1
.-u.
l l
o
~
g s
e
~
[W3/goasl n0L x gl/0 33
10-2 i
i i
g i
I I otal IO T
10-3 (aA)i + (ad)2 n
[
Y e
/
10-4 I
w.
- /
+d 10-5 Experimental Data l
l FREQUENCY, f (HERTZ) l 10-6 105 106 107 108 109 Figure 3:
Absorpton per Wavelength Versus Frequency f for 3.125 Narmal Cobalt-Polyphosphate Solution 5
'0 1
en icy l
G E
f
+
'0 o
1 n
D o
i
=
tu lo S
C su oeu q
A
'0 B
1 ro
+
F
)
A z
f tre y
H c
(
n f
eu Y
q C
e N
r E
F U
'0 s
Q u
8 1
E s
R r
F e
V 2
f
/a fo to l
'0 P
8 1
4 erag i
F 0
01 0
0 0
0 0
0 0
0 4
3 2
1 E2"oI_b.X lo
10-2 A+B:
'C aD+E p (Total) e it
/
pReaction 1 (fast) i 10-*
g 4
I I
pReaction 2 8 FViscous (slow) 10-*
I I
I I
i 10*
1 05 10*
10' 10*
10' i
I FREQUENCY f (Hertz) i l
Figure 5:
Absorption per Wavelength Versus Frequency f j
for Aqueous Glycine Solution i
l l
z H '0 G1 1
~
(
y n
cn o
e ti u
u q
b e
i r
r t
F no s
C u
s s
u r
o e
t c
i V
1 s
s H
a ami 2
FI V Mao f
i t
/
00 a
1 fo
)
n z
o tr i
e t
H a
(
l n uo f
w2 ci Y
l t n
lo m C
au SI Cl N
o E
eS U
t x
imb H '0 Q
ar me E
it R
F oW xa 2,
r e
t pe z
pn a
y Aa x M
t 1
do x r oe 0
ni r f 1
aD f
pi pD tp A
c n
ar d o xo nit EF aa t lu c c al 6
x a EC erug i
F zH*
M0 1
0 0
0 0
0 0
1 9
8 7
6 5
4
. 2m e ['hE I'
l lll l
l
10' 102 4 Viscous
=
p(Total Exact) h j Fast
/ Reaction
-e n 108 Reaction 1 2
h 10*
Reaction 2 P1 42 105 Range of Slow Experiments Reaction 10*
i i
i i
10' 10' 10' 10' 10' 10" Frequency f (Hertz)
Figure 7: Absorption per Wavelength Versus Frecuency f For p-01oxane Water Solution 38
rM ll l" l "pI
'l IN N
R E
I
),R U S. NUCLEAR REGUL ATORY COMMISSION somu 335 BIBLIOGRAPHIC DATA SHEET NUREG-0935 4 TlTLE AND SUBTIT LE (Add Volume No, ot appecorsatel 2 (Leave bianki /
Acoustic Wave Propagation in Fluids with Coupled
/
Chemical Reactions 3 aECiP s ACCESSION NO 7 AUTHOHIS) 6O E F.EPOR I COMPLE TED f ONTH YEAR Timothy S. Ma ulies and W. H. Schwarz
/
August 1984 9 PE RF OHMING OHGANI, TION N AME AND M AILING A DOHE SS tlac/u* lp Codel DALE REDOHT ISSUED Division of Risk nalysis and Operations l " ^a Office of Nurlear,egulatory Research August 1984 U.S. Nuclear T<egul ory Commission 6'"*"*"*'
'lashington, DC 205.
- 12. SPONSOHING OHGANI/ A flON, iME AND M AIL!NG ADDHESS //nou*
p cones Division of Risk Analy is and Operations Office of Nuclear Regu tory 9esearch 11 FIN NO.
U.S. Nuclear Regulatory ommis sir)n Washington, DC 20555 l
13 TYPt OF HE POH f et stoo cov t ut o unc. +ve d rest Technical Report SUPPL E VE N T AH Y NO f t 5 14 f t.ue tw a f 16 AHST H AC T (100 words or /eul This report presents a hydroac s ti theory which accounts for sound absorption and dispersion in a multicomponen mixtu of reacting fluids (assuming a set of first-order acoustic equations wi out diffu igon) such that several coupled reactions can occur simul taneously.
Gen al results af'g obtained in the form of a biquadratic characteristic equation (
lled the ,irchhbff-Langevin equation) for the cc.. plex propagation variable x=
( a + idc) in which a is the attenuation coefficient, c is the phase speed of t progressive wave and s.is the angular frequency.
Com-puter simulations of s nd absorption spectra haveQeen made for three different chemical systems each omprised of two-step chemicalKeactions using physico-chemical data availa e in the literature.
The relative chemical reaction and classical viscothe 1al contributions to the sound absorhtion are also presented.
Several discrepanc is that can arise when interpreting ult'rasonic measurements for estimating tnermo 'namic data (chemical reaction heats or volume changes) for multistep chemic reaction systems are discussed.
)
i r n e v e,oHos ANo o 1:ow N T AN At vsis
, i.. ut sucP r ous Ultrasonics /
Chemical Relaxation Wave Propagation Chemical Kinetics Physical Acoustics of Fluids Thermodynamics tth IDE Nit F IE HS OPE N E N Dt O TE H"S 18 AV AIL ABILI T Y S T A TE W N T P) SE LU HI T Y C L A55 f f" > ' 'p Uf/
21 NO OfPA:dN Ifnt l a ssi fiert Unli*ited 2" $E a"s"s'i f"i ed Uncl N RC 6 ORY 3 M sig su m
j r-UNITE 3 STATES souar eumssesu Zi C I NUCLEAR CECULATO7.Y COMMi.*JION
'037^',8 8, sits *zio r
g w
WASHINGTON, D.C. 3)S55 wasa 1 e mt n aur in O!
i 8)
OFFICIAL BU$1 NESS ptf4AkTV 50R Per/ ATE USE. SM Wt
?
i i
i D.
c ',
0:
C i-m.
do
$5
.):
<l m;
mm:
O' S.
L 1AN1RH 120555078877 LIS NRC y
.ADM-Dtv 0F TIDC BR-PDR NUREG
-.i P0t.!CY E PilB MGT 6
W-501 DC 20555 Z:
n F
E O.
m I
AI OO g
T' r"*m On ImE5>-
F1 33m
),
Od O
Z.
m
.j i
i r