Results and statistical analysis are presented for sound absorption in N2-H2O binary mixtures at room temperature. Experimental procedure, temperature effects, and preliminary results are presented for sound absorption in N2-H2O binary mixtures at elevated temperatures

Meredith, R. W.

Zuckerwar, A. J.

English

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jDEPARTMENT OF PHYSICS
SCHOOL OF SCIENCES AND HEALTH PROFESS,;ONS
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U THE CONTRIBUTION OF MOLECULAR RELAXATION IN
$, NITROGEN TO THE ABSORPTION OF SOUND IN THE
1.1J
C!^ ATMOSPHERE:'
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JJ R	 1 1	 :'
'. ww Allan J. Zuckerwar, Principal Investigator
8 and
Roger W. Meredith, Graduate Research Assistant
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Final Report_,
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.Prepared for the
National Aeronautics and Space Administration
Langley Research Center
Hampton, Virginia
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0 Under
Research Grant NSG I:Q4,
Harlan K. Holmes, Technical Monitor
Division
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Instrument Research
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April 1980
DEPARTMENT OF PHYSICS
SCHOOL OF SCIENCES AIJD HEALTH PROFESSIONS
OLD DOMINION UNIVERSITY
NORFOLK, VIRGINIA
Technical Report No. PTR 80-4
r
THE CONTRIBUTION OF MOLECULAR RELAXATION IN
NITROGEN TO THE ABSORPTION OF SOUND IN THE
ATMOSPHERE
By
r
Allan J. Zuckerwar, Principal Investigator
and
Roger W. Meredith, Graduate Research Assistant
Final Report
Prepared for the
National Aeronautics and Space Administration
Langley Research Center
Hampton, Virginia 23665
Under
Research Grant NSG 1324
Harlan K. Holmes, Technical Monitor
Instrument Research Division
Submitted by the
Old Dominion University Research Foundation
et	
P.O. Box 6369
 Norfolk, Virginia 23508
April 1980
[ b
-°J
THE CONTRIBU'ION OF MOLECULAR RELAXATION IN NITROGEN TO THE
ABSORPTION OF SOUND IN THE ATMOSPHERE
By
Allan J. Zuckerwar* and Roger W. Meredith**
SUMMARY
Work accomplished during the final two quarters of NSG 1324 falls into
two broad categories: first, sound absorption in N 2 -H 20 binary mixtures at
room temperature—changes in final results and statistical analysis; secondly,
sound absorption in Nz-H20 binary mixtures at elevated temperatures—experi-
mental procedure, temperature effects, and preliminary results. A summary
of publications resulting from research performed under this grant is
included.
SOUND ABSORPTION IN N 2 -H ZO BINARY MIXTURES AT ROOM TEMPERATURE
E:
F	 .
t	 ^
f
A. Changes in Final Results
A few changes regarding the evaluation of the da
Meeting of the Acoustical Society of America have led
of the experimentally determined relationship between
location of the N 2 vibrational relaxation peak on the
axis. 1 ) 2 Theoretical considerations suggest that the
linear:,
to presented at the 97th
to a slight modification
humidity h and the
frequency/pressure (f/P)
relationship should be
( f /P) 	 = ao + al h	 (Hz /atm)
	
(1)
* Research Associate Professor, Department of Physics, Old Dominion University,
Norfolk, Virginia 23508.
**Graduate Research Assistant, Department of Physics, Old Dominion University,
Norfolk, Virginia 23508.
S'"
N3t•
A weighted least-squares regression analysis yields the following updated
values for the constants:
aq	 0.013 i 0.012	 Hz/atm
al = 0,0200 * 0.0018
	
Hz/atm•ppm.
The water vapor content h is expressed in parts per rillion (ppm). The
limits of error represent the 95% confidence intervals for the random error
only, which was obtained by the computational procedure described below. The
systemat6c error, by far less significant, is still being formulated and is
expected to amount to less than i5% of the above values. The correlation
coefficient of the fit is 0.9938--which indicates an excellent linear fit.
The changes from the previous data include the addition of a new point
(at h 18 800 ppm), the correction of a numerical error (5237 ppm corrected
to 5327 ppm), the deletion of a point in which the dew point was not taken
immediately after the measurement, and improved rounding consistency.
B. Statistical Analysis
Figure 1 shows the sound absorption per unit wavelength p in N 2 vs.
(f/P) for four selected values of water vapor concentration. The solid lines
are the best fit of the experimental data points (circles) to the theoretical
Oebye relaxation curve:
( f/ P ) (f/P) max
max	 ( f/ p ) z + (f/P)max
The peak height 
µmax 
is known from thermodynamic considerations. The value
of (f/P) at which the crest of the peak occurs is called the "relaxation f/P"
and is designated by (f/P)max' the only adjustable parameter in Eq. (3).
Table I lists corresponding values of humidity and (f/P)max' together with
the 95% confidence intervals on (f/P) max , for ten sets of data taken daring
the course of the experiment.
The ten sets of data listed in Table I are shown in the plot of Fi g . 2.
The solid line is a plot of Eq. (1) with the best-fitted values of slope al
(2)
(3)
2
AID+''	 s	 ,..	 ^
f^
and intercept ao given in Eq. (2), The error bars are the 95% confidence
intervals on (f/P) max as listed in Table I, and, it is emphasized, include
only the random error.
The problem of computing the random error consists of two parts: first,
to determine the bounds on each (f/P)
max 
upon fitting the sound absorption
data to the theoretical curves given by Eq. (3); secondly, to determine the
bounds on ao and a, in the linear fit of Eq. (1). The statistical
distributions used here are explained in many standard texts on the
subject.3,405
In the procedure for determining the best value of (f/P)max to fit a
set of absorption data to Eq, (3), it is convenient to simplify notation
by using A 1/(f/P)
max
 as the adjustable parameter. Then, for the ith
point of a given set of data let u i (A) be the estimated value of absorp-
tion according to Eq. (3) and M the measured value of absorption. The
quantity S(A), the sum of the squares of the deviations, is minimized to
determine the value of A which gives the best fit:
S (A) =
	
	
[ u i (A) - Mi] 2
i
This is done numerically on an HP Model 9830A desk calculator. Initially
a rough estimate of A, designated as Ao, is obtained from the location
of the peak on a graph and S(A O ) is computed. Then A is changed by
small increments about A O and S(A) recomputed until it reaches a minimum
value. The corresponding value of A, designated by A, yields the best
fit and is determined to three significant figures.
The statistical problem may be stated as follows: How far can A deviate
from A before the difference between S(A) and S(A) becomes significant
at a given confidence level? This problem is resolved with the aid of the
"F distribution," which is designed to test the equality of two variances,
such as S(A) and S(A), and is suitable for small samples.*
*The variance is actually Eq. (4) divided by the number of samples, but the
F distribution may be applied to S(A) and S(A) directly since the number
of samples is the same for both.
r f
(4)
3
If S(A) is computed for an arbitrary value of A, the F distribution
is defined* such that
9 (A) 1 S (A) 1+ N P
- 
P F (P, N- P, 1- a)^	 (5)
Here P is the number of adjustable parameters, N the number of data points,
and 1 - a the confidence interval. In the present analysis P = 1 since A
is the only adjustable parameter, and I - a is taken to be 0.95, which corre-
sponds to twice the standard deviation of a Gaussian distribution. Knowing
P, N, and 1 - a, one can find F from available F distribution tables.
Then the entire right side of Eq. (5) is known. The parameter A is varied
until S(A) equals -the right side of Eq. (5). There are two values of A,
designated as Aa and Aa, which fulfill this requirement:
Aa<A<Aa	 (6)
The difference (Aa - Aa) represents the 1195% confidence interval" on A.
The error bound on Cf/P)max' defined here as
B 
= (Aa)-a. - (Aa)-1	 (7)
is an estimator of four times the standard deviation of A - ' = (f/P)max
Because S(A) is an asymmetric function of A, the bound is given as
B in Table I and not in the form ± 8(f/P)max'
Once the best values of A, thus (f/P) max , and their confidence inter-
vals are established, the ten sets of data of Table I are fitted to Eq. (1)
by a weighted linear regression analysis. The need for weighting stems
from the fact that the ten observations are not samples from the same parent
distribution, and the dependent variable—the (f/P) max I s—are characterized
by widely diverging variances. In accordance with established practice, the
data are weighted by the reciprocals of the variances, which in the present
analysis are represented by the estimator B:
*Eq. (5) is approximate because the theoretical relaxation curve, Eq, (3),
is nonlinear in (f/P) max , the quantity for which A -1 is an estimator.
4
-2	 (j = 1, 2, .., , 10).
s of as and a l are derived from the usual formulas:
(8)
(EwjYj)(wjxj I ) - (EWjxIYj)(EWjxj)
as _ .
	
	
,^:.	
,
(Ew,.) (F'w x. 2) - (9W „ ) 2
(EWj)(EW j xjy j )	 (Exj)(EYj)
al -	 ,
(Ew j ) (Ewj x j 2 )	 (EWjxj) 2
in which x  is the humidity h and yj the value of (f/P)max for the
jth set of data. The error limits on as and al are set equal to the
95% intervals of a "Student's t" distribution, which for small samples pro-
vides improved estimators for the mean and standard deviation. The distri-
bution is defined in the following equation:
<a>a±t	 s
c VN _- 2
where <a> is the true mean, a the sample mean, c the confidence level,
t  the confidence coefficient, s the sample standard deviation, and N
the number of samples or data sets. Values for t c , which depends only
upon c and N (more exactly, the number of degrees of freedom), are found
in readily available t distribution tables. In the present analysis a is
the best-fitted value of as or a,, and c is again chosen to be 0.95. The
standard errors so and s 1 , for ao and aa , respectively, are used to
represent s in Eq. (11):
1/2
( EW j (Y j - Yj)2](Ewjxj)2
50
j
1/2
'	 LWj(Yj — Yj)z
r	 s1
L
(9)
(10)
(11)
(12)
(13)
5
c' .z
w.xj	 Ewj
easured value of humidity h,
y
v, = measured value of ( f/P)max'
Y  = estimated value of (f/P) max' Eq* (1)
The factor (N - 2) appears in the denominator of Eq. (11) becausa the two
unknowns, slope and intercept, remove two degrees of freedom. For 8 degrees
of freedom t ,95 = 2.306. Upon substituting the data of Table I into Eqs. C8)
to (13) one obtains the best-fitted values and confidence intervals given in
Eq. (2) .
SOUND ABSORPTION MEASUREMENTS IN N 2 -H20 BINARY MIXTURES AT
ELEVATED TEMPERATURES
A. Experimental Procedure for Measurement of Sound Absorption at Elevated
Temperatures
The first part of the experimental procedure has remained the same as
before. The entire system is evacuated to 5 x 10 -5
 Torr, and if an impurity—
H 20 or CO 2 —is to be used, it is admitted to the system first. Nitrogen is
then admitted into the tube to a desired pressure. Mixtures of N 2 + H 2O are
allowed four days to homogenize to insure uniformity. A gas sample is taken
for independent analysis by gas chromatography.
The tube temperature is initially set at 298°K, a set of acoustical data
is taken, the pressure, dew point, and temperature are measured. The tempera-
ture is reset to 342°K, and the tube is left to stand overnight to come to
thermal equilibrium. The acoustical and ancillary data are obtained at the
new temperature, and this process is repeated again at 387°K. A second gas
sample is taken upon completion of the acoustical measurements.
There has been a slight modification in the procedure for measuring the
dew point with the Cambridge Systems Model 992 Dew Point Hygrometer. The
piping between the tube and hygrometer is evacuated to 5 x 10- 5 Torr prior
to being flushed with pure N 2 . Then the test gas is used as a final flushing
6
agent. The hygrometer has a feature allowing long-term measurement of dew
point by operating with the "auto balance" switch in the "on" position. It
has been determined that in the present experiments the precision of the
instrument is increased with the switch turned off. In addition, less gas
is needed for accurate readings at high values of humidity. This is impor-
tant because at low pressure the dew point was found to vary with presure.6
Finally, multiple dew point measurements of test gases are made before and
after the acoustical data are taken; the reproducibility of the dew point
measurement is within 1%.
B. Temperature Effects
Each time the tube temperature is changed the dew point of the N2-H2O
mixture also changes. At higher temperatures the water vapor previously
adsorbed onto the walls of the tube during the initial charging returns to
the vapor state when the wall is heated. Typical results for two different
mixtures are shown in Fig. 3.
The elevated temperatures also cause the broadband background noise of
the PCB quartz microphone to increase as much as fivefold. A water-cooling
system affixed to the microphone proved to be ineffective in reducing the
noise levels. The primary sources of the noise appear to be the increased
electronic noise of the microphone and increased thermal noise of the gas
inside the tube. Fortunately the length and quality of the acoustic decay
curves are not seriously affected.
C. Preliminary Results
A summary of the range of parameters--temperature, pressure, humidity--
and some corresponding values of (f/P)
max 
appear in Table II. In addition,
five sets of N 2 -0O 2 data were taken to provide background values for (f/P)
< 3 Hz/atm. A plot of three sets of data, one at.each selected temperature,
is shown in Fig. 4. Measurements at room temperature were taken for the
purpose of control. As before, the peak heights are known from thermo-
dynamics and the peak locations are determined from the best fit to Eq. (s).
Without exception all the data evaluated thus far indicate that the slope
a l in Eq. (1) decreases with increasing temperature— a surprising and
unexpected result with potentially far-reaching theoretical consequences.
7
eFor this reason more sets of data will be taken at elevated temperatures
than originally planned.
8
^ i
r
,
PUBLICATIOS
The measurements of sound absorption in Ito.,-H 20 mixtures, conducted under
NSG 1324, are the basis of a doctoral thesis and a:master's thesis, both in
preparation.
The following article has been accepted for pt rtF Ii.cation:
A.J. Zuckerwar and W.A. Griffin, "Resonant Tube for Measurement of
Sound Absorption it Gases at Low Frequency/Pressure Ratios," J. Acoust.
Soc, Am., scheduled For July 1980.
The manuscript for a sister article, "Effect of Water Vapor on Sound
Absorption in Nitrogen at Low Frequency/Pressure Rat li,os," is presently in
preparation. Additional articles covering the scope of the research grant.
are being planned, as is a (,,6^:,prehensive Contractor's Report.
The following are published abstracts of presentations made at professional
society meetings:
1, W.A. Griffin, "A4^gon Calibration Study of a Resonance Tube at Low
f/P Ration," Virginia Academy of Science, 56th Annual Meeting, May 1978.
2. A.J. Zuckerwar and W.A. Griffin, "Resonant Tube for Measurement
of Sound Absorption in Gasses at Low f/P Ratios," J. Acoust, Soc. Am, 64,
Suppl. No, 1, S61, 1978,
3. W.A. Griffin and A.J. Zuckerwar, "Location of the Nitrogen Vibrational
Relaxation Peak on the Frequency/Pressure Axis as a Function of Humidity in
N2 -H 20 Mixtures," J. Acoust. Soc. Am. 65, Suppl, No. 1, S48, 1979.
4. A.J. Zuckerwar, "Radiation Losses in Resonant Tubes," J. Acoust,
Soc. Am, 65, Suppl. No. 1, S75, 1979.
9
,'	 i
RED ERRNCES
1, W.A. Griffin and A.Jb Zuckerwar, "Location of the Nitrogen Vibrational
Relaxation. Peak on the Frequency,'Pressuxo Axis as a Function of Humidity
in N2 -H20 Mixtures," J. Acoust. Soc. Am. 65, Suppt. No. 1, S48 (1979),
2. A.J. Zuckerwar and W.A, Griffin, Technical Report P'Ot-J9-6, Old Dominion
University, Department of Physic:, August 1979.
3. N.R. Draper and H. Smith, Applied Regression Ana lysis, Wiley, 1966.
4. K. Rektorys, Editor, Survey of npi)licable Mathematics, MIT Press, 1969.
S. P.G, Hoel, Introduction to Mathematical Statistics, Wiley, 1962.
6. See reference 2,
tr I
10
r- - - - - -,--I 'im
G 1. R+olaxaton Cf/P)1nax vs. Humidity in Sound Absorption
Measurements in N2-H20 Binary Mixtures at Room
Temperature (2970X)
Data Humidity h ( f/P)max Bounds an
Set (ppm) (Hz/atin) (f/P)max
1 2,5 0.063 0.058 - 0.070
2 57 1.18 1.06 =	 1.31
3 167 3.03 2.43	 3.66
4 338 6.17 5.62	 6.76
5 838 2012 18.0	 - 22.6
6 4112 96.2 69.4 - 129
7 5327 97.1 62.5 -	 143
8 9026 196 133 - 28
9 14 400 304 231 = 402
10 18 800 435 321 - 588
a 95% confidence interval using the F distribution.
11
Table II. Range of Data and Preliminary Results for Sound
Absorption Measurements in N 2 -H 2O Binary Mixtures
at Elevated Temperatures
Temperature (f/P)maxPressure Humidity
(°K) (atm•abs) w	 (ppm) (Hz/atm)
298 1.0 18 500 a
1.0 7410 a
35.4 110.1 2.11
35.4 17.1 0.267
80.5 s1.5 0.046
342 1.2 21 840 a
12 8770 a
} 43.7 84.8 1.23
43.7,
	
89.4 <1.5 0.0327
387 1.3 24 710 a
1.3 10 430 a
53.4 217.1 2.10
53.4,
	
105.5 10.9 0.175
a Sound absorption data not yet evaluated.
wy
s i
r	 1
F
12
ti
2
1
0
3
rn
(U
as
 2
w 0
a
z
0
0
U)
a3
0
Z 0
M
0
U) 3
2
0.1	 1	 10	 10 
2
	
103
FREQUENCY/ PRESSURE (Hz/atm)
Figure 1. Sound absorption versus frequency/pressure ratio for four selected
N2-H20 binary mixtures at 297°K. The solid lines are best fits
to Eq. (3).
J	 .I
c,	
I	 10	 102	 103	 104
	
105
H^0 MOLE FRAC - ION (ppm)
Figure 2. Relaxation frequency/pressure ratio (f/P)max versus humidity h
in N Z -Hz0 binary mixtures at 297 0 K. The error bars designate
95% confidence intervals.
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The contribution of molecular relaxation in nitrogen to the absorption of sound in the atmosphere

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