Acoustic determination of the critical point of methane utilizing a variable volume spherical resonator under isothermal...

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Title:
Acoustic determination of the critical point of methane utilizing a variable volume spherical resonator under isothermal conditions
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xii, 100 leaves : ill. ; 29 cm.
Language:
English
Creator:
Rentz, D. C., 1957-
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Notes

Thesis:
Thesis (Ph. D.)--University of Florida, 1994.
Bibliography:
Includes bibliographical references (leaves 95-99).
Statement of Responsibility:
by D.C. Rentz.
General Note:
Typescript.
General Note:
Vita.

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University of Florida
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All applicable rights reserved by the source institution and holding location.
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Full Text








i



ACOUSTIC DETERMINATION OF THE CRITICAL POINT OF METHANE
UTILIZING A VARIABLE VOLUME SPHERICAL RESONATOR
UNDER ISOTHERMAL CONDITIONS



















By


RENTZ


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
flT- mrTTr TT TrTlnf lTmr m7 r' r'T flr1l7\ Ti Tia fln mTrTfl T rTTYT rTT T T.NLATFTm


































Copyright

by


1994


Rentz






































To Ashley,


believed


could


anything.
















ACKNOWLEDGEMENTS


impossible


acknowledge


people


have


helped


production


this


work.


However,


there


are


several


teachers


have


been


instrumental


in propelling me


completion


my graduate


studies.


There


my wife,


Ashley


Rentz,


taught


me of


matters


heart.


My parents,


and Marilyn


Rentz,


have


taught


me how to


live


among


others.


friends,


Patti


and Pierre


Robitaille,


who have always been


the example of how


would wish


to treat


others.


My mentor,


Samuel


Colgate,


who has


taught me how


work


out


within


above


craft.


rest.


There


those


are


others,


mentioned


these


those


that


stand


have


gone


unsung,


thank


you.
















TABLE


OF CONTENTS


ACKNOWLEDGEMENTS


. . IV


LIST


OF TABLES


. a a 1Vll


LIST


OF FIGURES


Viii


ABSTRACT


CHAPTER


INTRODUCTION


a S a aXl


Motivation


Examining


Critical


Point


Methane


. 1


CHAPTER


HISTORICAL


OVERVIEW


. a. 5


First Thought
Background
Critical


point


action


from


sotherm


studies


Critical


point


location


from


sochore


studies


Criti
Speed
Point


point
Sound


location


from i


Probe


sobar


the


studies
Critic


CHAPTER


THEORY


Wave
Acou


Mechanics


stics


Acoustic


Spectrum Analys


CHAPTER


EXPERIMENTAL


Procedure
Initial


Attainment
Data Acquisition
Calibration


Loading
f System


* a a a
S S S


Criti


* a a a a a a
* a a a a a a
* S C a p 5 S a


Density


a a a a a a a a a a a


S1










48


Transducers


System RTD Placement . . .
Bellows Construction Detail


CHAPTER


RESULTS


Spherical Resonator
Acoustic Spectra
Data Manipulation
Curve Fitting


* a a a
* a a


Critical pressure
Critical temperature
Thermodynamic Surface


* a a a a a a a
a a a a a S a a
* a a a a a a a a a


CHAPTER 6


DISCUSSION
The Critical Point of Methane


* a a a S a a a
* a a a a a a


Gas Impurity
System RTD


Heis


Pressure Gauge


Precision of Technique


Personal Note

REFERENCE LIST


BIOGRAPHICAL SKETCH
















LIST OF TABLES


Table


3-1. ei, eigenvalues
function of order m.


spherical


Besse
* S t


Table 4-1.


Instrumentation Summary.


S54


Table 4-1.


Table 4-1.


(continued)


(continued)


55


56


Table


5-1.


System


dependent


variables


fitted to experimental data.


equation
. 80


Table


6-1.


Methane critical point parameters


found in


the literature.
















LIST OF FIGURES


Figure


2-1.


Typical P,


and T


associated two dimensional pi


surface construct
ots. .


Figure


2-2.


Pressure-volume isotherms of carbon dioxide


in the critical region.


Figure


2-3.


Pressure-temperature


isochores


in the critical region.


of benzene
.* .


Figure


2-4.


Determination


critical


point


hydrogen from measurements of the liquid density pi
and gaseous density p . . .


Figure


3-1.


Acoustic


spectrum


methane


from


spherical cavity,


(three inch radius)


Figure


3-2.


Typical detail of individual gas resonance


frequencies,


(excerpts from fig.


3-1)


Figure 4-1.


Figure 4-2.


Experimental apparatus schematic.


Control instrumentation schematic.


Figure


4-3.


Calibration


plot


the Heise gauge


vs.


ADC units.


. . 42


Figure 4-4.


Pressure calibration setup.


Figure


4-5.


Response


RTD under identical


system RTD
conditions.


and precalibrated
S. 44


Figure


4-6.


Plot


exhibiting


System RTD


strong


linear


Precalibrated RTD


response


temperatures.


Figure 4-7.


Triple point


of water calibration


for the


precalibrated RTD.


4 n.. -.a 0 1 rnetf rk nm LI t


45


46


n7


u


r?:,,,,, /1_0









Figure 4-10.
shroud.


Figure 4-11.


Construction detail of system temperature


Construction detail of bellows assembly.


Figure


5-1.


Typical


acoustic


spectrum


recorded


well


away from the critical point.


S. .59


Figure


5-2.


Typical acousti


critical point.


spectrum recorded near the
S . . 60


Figure


5-3.


Excerpt from an acousti


spectrum near the


critical point.


Figure


5-4.


time


Excerpt
volume


from
has


acoustic


changed,


spectrum


(frequencies


after
are


shifted toward zero frequency)


Fiaure


5-5.


Superposition


two


acoustic


spectra


excerpts
frequency


illustrating the rapid change in resonant


amplitude


that


accompanies


small


change


in volume near the critical point.


Figure
of


5-6.


Typical


sound


relationship between system speed


system


volume


near


critical


temperature.


Figure


5-7.


Typical


relationship


between


speed of sound and the system density.


the system
S . 65


Figure


5-8.


Typical


relationship


between


system


speed of sound and the system pressure


Figure 5-9.


Plot of all data sets of pressure and speed


of sound on a common set of axes.


Figure 5-10.
data run.


Largest temperature variation for a


single


Figure


5-11.


Pressure


dependence


temperature


data


run


that


deviates


most


from


isothermal


conditions.


S73 73


Figure


5-12.


Frequency


dependence


temperature


data


set


that


deviates


most


from


isothermal


conditions.


, in, ,-r~ 1Q n-I .-~4- -11 r11 A~r4- n vrerFrtr0 -a er CaA


63


64


c; nrrrh E,17


~rrn C nm innn~ nF


nlnt nC









Figure


5-14.


variables;


Three c
pressure,


dimensional


temperature


plot


and


c system
speed of


sound.


Figure


5-15


system

Figure 5-16.


Plot
pressure


Plot


system


speed


sound


versus


with fitted equation overlaid.


system temperature versus system


pressure corresponding to minimum values for system


speed


sound for each data run.


Figure


point


7. Thermodynamic
of methane.


surface


critical


Figure 6-1.


Sub critical pressure data taken near the


critical density.


. .a 89


Figure


6-2.


Above critical pressure data taken near the


critical density.


Figure 6-3.


Sound velocity at selected temperatures


a function of pressure


Figure


6-4.


Graph


sound velocity


function


pressure containing all twenty data runs from this


experiment.


83


92















Abstract


of Dissertation


:he University
Requirements j


ACOUSTIC


UTILIZING A


of Florida


Degree


DETERMINATION


VARIABLE VOLUME


ISOTHERMAL


Presented


in Partial
!e of Doctor


the Graduate


School


Fulfillment of
r of Philosophy


CRITICAL POINT


OF METHANE


SPHERICAL RESONATOR


UNDER


CONDITIONS


Rent


April


1994


Chairperson:


Major


Samuel


Department:


Colgate


Chemistry


novel


apparatus


study


thermodynamic


properties


fluids


been


fabricated


utilized


examine


critical


point


of methane.


A spherical


acoustic


resonator


assembly


with


variable


volume


was


used


sensitive


probe


examining


thermodynamic


surface


methane


critical


region.


resonator


was


built


operate


over


broad


ranges


pressure


and


temperature


4000psi


77-525K)


unique


stainless


steel


bellows


arrangement was incorporated to allow a variable system volume


which


first


time


permitted


acoustic


phase


mapping


under


isothermal


conditions.


frequency


third


radially


symmetric


normal


mode


vibration


was


measured and









measured


reported


4.60810. 0015MPa


189.695z0.021K


respectively.


plot


system


pressure,


temperature


sonic


speed


was


constructed


delineate


thermodynamic


surface


critical


region


of methane.
















CHAPTER


INTRODUCTION


Motivation


Examining


the Critical


Point


of Methane


Methane


principal


component


natural


gas.


such,


used


extensively


fuel


many


different


processes.


demand


this


natural


resource


established methane


a commodity


of great


economic


as well


as political


importance.


As a result


of this demand,


methane


has been made


available on an increasing scale


throughout


twentieth century.


This availability and importance have made


methane


attractive


material


scientific


study.


Consequently


, a wealth


of summarized data can be


found in


scientific


literature.


ready


methane


unified


equation


availability


made


numerous


appealing


state.


tabulations


theoretical


equation


data


work


state


mathematical


model


system


that


relates


pressure,


temperature


specific


volume


density


analytical


way.


history


efforts


to establish


reliable


equations


of state


rich dating back


to the work


of Boyle


as early


nr'm~ ~ r t- I ma~ mt~ rc~ nhnt h~7 bcP c'


4nt rndiinrr


1 G; Gt 1


nrn


th rr t


time


I











No general


analytic


density.


equation of


practical


This


state,


values


been


however,


pressure,


particularly


yet proven to


temperature


detrimental


equations


exist


which


are


reasonable


agreement


with


experiment


under


limited


conditions


that


are


toward the extremes


of the


three


variables.


Since


great


majority


scientific


activity


involves


matter


states


which


are


removed


from


outermost


conditions,


major


efforts


toward


achieving


more


unified


equation


state


slowed


to a crawl


many


years.


Recent


developments


chemistry,


physics


and


astronomy


have


involved


studies


matter under


conditions


well


beyond


those


addressed


established


equations


state.


Those


temperatures and pressure


exhibited by


laser induced plasmas


observable


celestial


bodies


were


accessible


time


when


traditional


equations


state


were


being


formulated.


Consequently,


new


equations


applicable


over


broader


ranges


are


needed,


activity


generating


such


equations


growing.


critical


region


proven


especially


difficult


characterize analytically because it exhibits unusually


large


fluctuations


in the


local


density.


Typically,


a gas


is much


less


dense


than


its


corresponding


liquid.


critical












vapor molecules


tend


aggregate


small


clusters.


Some


thermodynamic


relationships


become


discontinuous


critical point.


The ability to describe such behavior is very


difficult mathematically.

The most fruitful treatment of discontinuities associated


with


critical


region


utilizes


scaling


laws


link


thermodynamic behavior on either side of the critical point.

Thermodynamic properties on one side of the critical point are


dealt


with


almost


separate


entities


from


same


properties on the other side.

in thermodynamic behavior, i


Because of this drastic change


t has become common practice to


base


equation


state


formulations


beginning


critical point.


Thermodynamic detail is then viewed in terms


of its distance from the critical point.


Mathematically,


extensive


use


made


reduced


variables such


as4'o


TI~


- T-T_


~red


P-Pc

Pc


and


P led


P-Pc
Pc


where


temperature


, P


pressure,


density


and


symbols


with


subscript


c represent


same variables at the critical point.


It is apparent that for


any theoretical model based on this approach to be successful,


r. ~ ~ ~ ~ ~ ~ ~ ~ n 4-1, rr t; r naa iI


- 4 r .-.-


I


n rr; ~?1


tkl~


Ckn n~r~mnt~r cl


%--


Ir ~u











methane.


Such a


question calls


a reexamination of


critical point parameters for methane.
















CHAPTER


HISTORICAL OVERVIEW


First


Thoughts


In order to put the current work


in proper context,


it is


important


acquainted


with


past


efforts


examine


critical region and identify the critical point


The relative


merits


and drawbacks of


eac


h method must be evaluated in order


illustrate


novelty


and


utility


work


presented


here.


Ideally


current


work


should


represent


advance


in experimental


technique and yi


eld result


of high precision


accuracy.


Final


judgement


extent


current


work achieves these goals


left


to the reader,


it is the


burden


author


illustrate


they


were


pursued and


what


been


accomplished.


Background


single


component


system


critical


point


typically


identified


some


type


phase


diagram.


pressure,


molar


volume


temperature


relationship


can


represented


a three


dimensional


construct.


surface


1


-. *


C.~~~~~ *: -n-r '*l~ -F' -rN


-4 h D m n


~n rC i nn


II


r~rn I-









6

the highest temperature on the coexistence line and is defined


critical


point.


Figure 2-1. Typical


and


surface


construct


associated


two dimensional


plots.


three


dimensional


construct


figure


cumbersome


projections


difficult


figure


visualize.


represents


Each


simpler


three


view


thermodynamic


surface.


upper


left


view


shows


versus


plotted


various


values


constant


volume.


Each


line











data


taken


a unique


constant


temperature


termed


isotherm.


Similarly,


lower


center


view


data


plotted


while


maintaining


constant


pressure.


Each


line


represents data


taken at


a different


constant


pressure and is


termed an isobar.


Any of the three projections has suffic


lent


information


reconstruct


three


dimensional


surface.


Several


established methods


for determining the location


critical


point


exist.


Traditional


methods


correspond


to production of one of the three projections mentioned above.

A short discussion of each is illustrative of the difficulties


faced


experimentalist


yields


insight


into


problems


that


must


addressed


achieve


accurate


measurement


critical


point.


Critical


point


location


from


isotherm


studies


This method requires the temperature to


be held constant


while


molar


volume


the density)


slowly


varied.


series


pressure


vs.


volume


plots


2 IS


reproduced


figure


change


2-2.


becomes


molar


depressed


volume


and eventually


varied,


reaches


pressure


a stationary


value


as the critical point


approached and surpassed.


8 The


slope


of the


isotherm


(aP/av


T approaches


and becomes


zero


and below


critical


point.









8

the critical point it is quite apparent that the slope of the


line never reaches


zero


over


a range of


volumes.


Likewise,


well below the critical point it is apparent that the slope of


the line


zero over a range of volumes.


However,


near the


critical point


, it becomes a subjective matter to determine at


what values of the state variables the slope of the line first


becomes


zero


and not


just merely close to zero.


Figure


2-2.


Pressure-volume


isotherms of


carbon dioxide


the critical region.





Critical point location from isochore studies


75.0 ---- -- ..-- --

742 ---- ------ ...- --- ---


73.8
Z 734. ,.o-.o.. ..,..,, _



;,.2-------F----^^ -------
B\
'I12


71.8----

71.6 _-l-- _-----3-----------29S-


31 35 39 43 47 51 55 59 63 67
v, x 104 = volume (amagat units) x 104











loaded


with


predetermined


density


material


under


study.


temperature


varied


pressure measured.


plot


pressure


vs.


temperature


constructed.


average


density


within


containment


vessel


lower


than


critical


density,


then


a downward


deflection


observed


graphed


data.


onset


deflection


occurs


corresponding


point.


Conversely,


average


density


within


containment


vessel


higher


than


criti


cal density,


then an upward deflection is observed in the


graphed


data


precisely


corresponding


bubble


point.


Only when the average density is equal to the critical density


does


deflection


occur.


Figure


illustrates


typical


results


method


benzene.


This


technique


straightforward


perform


, but


precision


density


accuracy


required


are

even


poor.

a modest


Substantial


break


change


or discontinuity


plot.


Even


use


curve


fitting


differentiation


techniques


still


fails


yield


high


precision.

The second method dealing with isochores is equivalent to


the first,


but takes advantage of


a peculiarity that occurs at


critical


point,


namely,


disappearance


liquid/gas


separator,


meniscus


critical


point,







































Figure 2
critical


3. Pressure-temperature isochores of benzene
region.


in the


disappears


middle


containment


vessel


uniform


horizontal


cross


section.


average


density


system


less


than


critical


density,


meniscus


will


appear to fall

with gas. If


slowly and eventually the vessel


on the other hand,


will be filled


the average system density


greater


than


critical


density,


meniscus


will


slowly


rise


eventually


vessel


will


filled


with


liquid.


temperature,


pressure


density


which


meniscus


disappears


precisely


halfway


vessel


are


taken


62 -
S CURVE 4 _____
____9 -5- -i I ---
I _BENZENE ____,______
I } ^ 70 cc CHA GE I s
-
2 c c i30b CAGE _____ ,:
- 1500= t50 cc CHA GEI _____ __
46
-- 38 --1504 ce CHAC-E "-


S34 I

30 0 I
f26 --


I8r
14, -I-
10 '-
215 220 225 230 235 240 245 250 255 280 265 270 275 280 285 290 295 300 305 310
Temperature C











critical


point,


however,


difference


between


thermodynamic properties of the liquid and gas becomes minute.


Therefore,

a result,


the meniscus becomes more difficult to observe. As

the reported values of the critical point parameters


eventually become dependant on the subjective


judgement of the


experimentalist.


Critical


point


location


from


isobar


studies


This


technique


known


method


rectilinear


diameter


plotting


illustrated


densities,


figure


(related


11,12


molar


requires


volume),


liquid


saturated


vapor


over


wide


range


temperatures


extending


close


possible


critical


point.


average


densities


calculated


also


plotted


on the


same


graph.


The average density


and the temperature are


fitted to an


equation

extended


form


until


2(pq+P'


=po+aT.


intersects


analytical


coexistence


line


curve.


values


obtained


from


intersection


are


taken


critical


point


parameters.


density


liquid


change


very


rapidly


near the critical


point and defining the shape of


the density


curves


becomes


very


difficult.


Also,


line


representing












more


than


reasonable


estimate


critical


point


parameters


this


method.


Figure 2
hydrogen
gaseous


from


Determination
measurements


critical


liquid


point


density


density


Speed


Sound as


a Probe


the Critical


Point


Each


above


mentioned


techniques


probing


critical


point


requires


certain


amount


subjectivity.


Considering


measurement


isotherms,


example,


ability


distinguish


between


line


slopes


almost


equal


zero


actually


equal


zero


typically


subject


nvn;mn 1nrna ni- -~ rxrr rrrh4av +-h 1n +ha 1 n4


0.08

0.07

0.06

0.05

0.04

0.03

0.02

0.01

0.00


-258 -256 -254 -252 -250 -248 -246 -244 -242 -240
T(C)--


n f: rlnra rt~; nt~~ tl; th


L^











accuracy.


Examination


projection


figure


corresponding


isotherms


P-V


plane


shows


critical


isotherm


passes


through


point


where


isotherm


slope


first


becomes


equal


zero.


A function


which


depends


on the


slope of


the critical


isotherm can have


a very rapidly


changing value at


point


dependent


slope of


T and


will


such a


also


function will


zero


critical


point.


Consequently,


such a


function


exists


would have


rapidly


changing


slope


that


goes


zero


critical point

in the critical


may be more


instead of a very slowly changing slope


isotherm in figure 2-1.


readily


as seen


This type of behavior


identifiable experimentally than deciding


when


virtually


flat


line


actually


becomes


completely


flat.


Thermodynami


explores


many


relationships


between


state


variables.


13 These


include


density,


enthalpy,


the free energy and others.


Heat capacity


is such a variable.


One equation


that


relates


heat


capacity


system at


constant


volume


to other


state


variables


av 8v
r~s arp


Where


is the molar heat capacity
Cis the system entropy
S is the system entropy


ap
9^


ap2/av











Since


approaches


and


becomes


zero


critical


point,


while


other


terms


have


nonzero


finite


values,


heat


capacity


critical


state


also


becomes


zero.


A three-dimensional plot of this state function against


temperature


molar


volume


would


observed


a spike


depression


that


extends


asymptotically


towards


zero


from


relatively


gently


curved


surface.


This


type


behavior


very


well


suited


locating


critical


point


thermodynamic


system,


spike


depression


much


more


readily


observable


than


the change


in slope of


an almost


flat


line.


Unfortunately,


accurately measuring the heat capacity for


a system is an arduous task.


What


is needed now is a property


with similar behavior but one which is easily measured.


Such


a state


function


propagation


speed


a pressure


wave


through


system


speed


sound.


thermodynamic


equation


speed


sound i


V M


lp-
Byv


Eqn.


this equation,


7 is


ratio


heat


capacity at


constant


pressure to


the heat


capacity


at constant


volume and


Molecular


weight


specie


s under


study.


1!5


(ap/av


ap











should


very


sensitive


probe


critical


point.


Several methods are known


for measurement


of the sonic


speed.


The method


employed


this


work


is extremely


sensitive.


involves measurement


of standing acoustic wave


frequencies of


cavity


bound


systems.


Virtually


type


containment


vessel


thermodynamic


well


fluid


characterized and


used


wave


long


equation


geometry


solvable


allowed


cavity


resonator


resonant


been


utilized


frequencies.


with


Often


sound


a cylindrical


generator


sound


receiver


located


opposite


ends


cylinder.


However,


propagation of sound through a cylinder is subject to


a bothersome perturbation


to viscous drag on the


cylinder


walls


which


complicates


final


analysis.


more


satisfactory


technique


utilization


spherical


resonator


that


accommodates


some


purely


radial


modes


vibration.


Such


vibrations


correspond


sound


wave


emanating from the center of the sphere and travelling outward


symmetrically to


the chamber wall


and rebounding back toward


the center.


These radial modes


are


free


from viscous effects


which


arise


when


sound


waves


have


components


motion


laterally


along


vessel


walls.


more


detailed


look


theory of


spherical acoustic resonators and their use is found















CHAPTER
THEORY


Wave


Mechanics


transmission


wave


through


fluid


medium


governed by


general


wave


equation18


V2T


Eqn.


Where


Laplacian,


the wave


function


time


wave


velocity.


Laplacian


spherical


coordinates


= --r-
r2 arr


9+
8r


1 a .
r1n sin6e +
r 2sine d 6


1 82
r2sin2O a2


wave


equation


assumed


standing


wave,


that


wave


whose


amplitude


does


depend


time,


then


possible


assume


separable


wave


function


such


that


SWe-ieEt
Eqn.


Where


= -1,


= 2xv,


= frequency.


azlY












Substitution


eqn.


into


eqn.


yields


v2w


r-2
= G
C2


Eqn.


Further,


each


assumed that


spherical


the wave


coordinates,


function


then


is separable


equation


spatial


wave


function


can be


written


= R(r


9(0) (4, )


Eqn.


Substitution of


eqn.


into eqn.


and utilizing the


Laplacian


yields


1 1 d 2dR(r)
R(r) r2 dr dr


1 1 1 d de(0)
(9) r2 sin dOsi dO
8(0) r2 sin0 de de


1
0(,


Eqn.


d2 ()


r2Sin28


task


of finding the wave


function


that


satisfies


eqn.


daunting


appearance.


However,


possible


separate eqn.


into


three


factors each of which deals with


only


one


three


spherical


coordinates.


Once


this


accomplished,


each


factor


separately


zero


rae~' 1 ~ C' fl A-i mane i nn 2 1 ffo arot orln ; n n YC


c2
C2


~ n 1 tr a r"j


~;mnnc;nn~l


rn ~r r 1 t ; n n


P~11=lt inne


;1 rP


/-\Tn












d 2 dR(r)
dr dr


+( 2 2r-A)R(r)
c2


Eqn.


sind sine dO (0)
dO d6


d20 ($)


+ (Asin2e -B) 8 (8)


+B0 (4)


Eqn.


Eqn.


Where


are


separation


constants.


Equation


represents


axial


angle


contribution


the overall


wave


function and


the easiest


to solve.


allow


separation


constant


equal


, then


general


solution


would be


= sin(n )


cos (nt)


Eqn.


Since


each


these


functions


periodic


boundary


condition requires they be single valued


as phi changes by any


multiple


Therefore,


separation


constant


constrained


have


integral


value.


That


Substitution


eqn.


rewriting


yields


sind sine dO
dO d6


+ (Asin2 -n2) (8)


(9,











Since


separation


constant


squared,


new


information


values.


gained


Making the change of


considering


variable,


negative


= cos


integer


yields the


following


(1-12) d2)


dO (ii)


-'IA-


n2
1-T12


6(n)


Eqn.


3-10


Eqn.


3-10


recognizable


as a Legendre function,


provided the


separation


constant


equal


m(m+l)


and


constrained


integer


equal


greater


than


zero.


notation


Legendre


function


written


because


solution


differential


equation


assumed


to be


a polynomial.


general


solution


eqn.


3-10


ecrl


= Pm(r1)


= (1-2)n/2 dn Pm()
dn P


Eqn.


3-11


where P,


the associated Legendre polynomial derived from


following


recursion


formulas


(-m) (1+m) 2
-' .+


(-m) (2-m) (1+m) (3+m


TI4 +.'.


+ (-m)(2-m)(4-m)-.(2s-2-m) (1+m) (3+m) .-(2s-l+m) 28z+..
2s!


= a[l+


P," (rl


Pm











For m


P, (rl)


(1-m) (2+m) 2+
3!


(1-m) (3-m) (2+m) (4+m) 4+..
5!


+ (1-m) (3-m) -(2s-1-m) (2+m) (4+m)---(2s+m) 2s+...
(2s+l) !


value


chosen


also


equal


highest


value


power


independent


variable


found


Legendre


polynomial


obtained


from


above


recursion


formulas.


Consequently,


chosen


such


that


less


than


then


an


From this,


it follows that the general solution stated in eqn.


3-11


additional


constraint


that


0

The combination of


eqn.


and 3-11


yields


the solution


angular


portion


overall


wave


equation.


Traditionally,


this


combination


functions


been


termed


spherical


harmonics,


Ym.n (1,) ,


takes


form


=e(Ti) (


=(1-12) 2


Eqn.
dan COS
dn ()sin(n)
nr


3-12


arl[l


Y,. n(rl











Substituting


= m(m+l)


into


eqn.


yields


d 2 dR(r)
dr dr


+ (k2r 2-m (m+1)


R(r)


Where k


Expanding


derivative


and


making


change


variable


r=kr


leads


d 2R(()


2 dR(C)
C dC


+[2 -m (m+l) R()
C2


Eqn.


3-13


This


equation


s a


general


solution


R(r)


known


spherical


Hankel


function


of order m,


m, ()


Rm (C)


-= h,(C)


combination


= i-m (m+s) !
i( s! (m-s)!


eqns.


3-2,


3-12


.
2(


3-14


Eqn.


3-14


gives


overall


solution


wave


function


defined


eqn.


=Rm (C)Ym,(n (i, )e- t


n
iC

(1i12 )2


m+s) 1


(m-s


dii


I i
2C I


Eqn.


3-15


sin (C') e- it


Acoustics











most


easily


related


thermodynamic


variables.


constraints


wave


function


provide


bridge


between


theory


practicality.


first


constraint


is that


the sound introduced inside


spherical


cavity


allowed


propagate


throughout


cavity.


spherical


Hankel


function,


eqn.


can


separated into two portions;


a real part and an imaginary part


follows


real


part


this


separated


function,


j,((),


known


spherical


Bessel


function


order


while


imaginary


part,


For


is known as a


example,


spherical


first


Neuman


spherical


function


Hankel


of order


functions


corresponding to


and m=l


are


Ro )


R (C)


e iC


e i
-i
C2


substituting


identity


ei =cos


and


rearranging


yields


= ho(C)


= 0 (C) +ino (C)


SEan


3-16


C


j,(C)+in, (C)


h, (C)


n, (r)


(r)+isin(r)


Y\












Ri (C)


= j ((C)+in,()


Ssin(()
C2


cos(C) -sin(C)
c C -- -


Scos(C)
C2


Eqn


3-17


While


either


real


part,


, or


imaginary


part,


are


solutions


radial


contribution


wave


equation,


only


real


part


also


property


of having


finite

just


Once


value

j ( ) :


first


when


standing


spherical


Therefore,


waves


inside


Bessel


can be


an enclose


functions


are


simplified

d chamber.


known,


subsequent


function


j,(


can


derived


from


recursion


formula


= C (2m+l


C) -j (,m-) (c)


second constraint


is that


as the acoustic wave meets


wall


chamber


must


cease


forward


motion.


Writing this


constraint


in mathematical


terms


leads


dRm(()
dr ra


inside
dj m (() resonator
dr r=a


Eqn.


3-18


where a is


radius of


spherical


cavitv. A


Lpplyinq this


nm(U),


h, (C)


R,(r


L^ A


-- A













dj
d


C inside
() resonator
r=a


= Tan(C)-C


Eqn.


3-19


dj1
d


\ inside
1) resonator
Ira


sin(C)


+ 2cos (C)


2sin(C)


Eqn.


3-20


There are an


infinite number of


solutions or eigenvalues


that


satisfy


either


eqns.


3-19


3-20


Separate


eigenvalues


are


designated by


general


symbol


5t,mrn


where


represents


where


infinite


series


solutions


particular


solution


falls.


analogous


manner


naming


more


familiar


hydrogen


atomic


orbitals,


termed


eigenvalue,


3 iS


termed


eigenvalue


forth.


owest


twenty-three


eigenvalues


were


calculated


Lord Rayl


eigh


187221


and H.


Ferris


extended


list


include


lowest


eighty-four


eigenvalues.


22 The


list


recreated


table


Each


eigenvalue


represents


distinct


normal


mode


acoustic vibration of


an elastic


fluid constrained by a rigid


spherical


wall.


Drawing discussion results together


leads


= kr r


Eqn.


3-21


= oa
C


2-91i-,c


1- rn ntr hay


'.4h~ I~ II i


fln1li nf fnf


2tra
V


nn


c


IIL i. I


I I I -


|











Table
order


3-1.


U( -,


eigenvalues


spherical


Bessel


function


.08158


Name


.4079
.9473


Name


4.49341 1 0 is 18.1276 2 11 2n
4.51408 1 3 if 18.3565 3 8 3k
5.64670 1 4 ig 18.4527 1 16 lu
5.94036 2 1 2p 18.4682 5 3 5f
6.75643 1 5 lh 18.7428 6 1 6P
7.28990 2 2 2d 19.2628 4 6 4i
7.72523 2 0 2s 19.2704 2 12 2o
7.85107 1 6 li 19.4964 1 17 iv
8.58367 2 3 2f 19.5819 3 9 31
8.93489 1 7 lj 19.8625 5 4 5g
9.20586 3 1 3p 20.2219 6 2 6d
9.84043 2 4 2g 20.3714 6 0 6s
10.0102 1 8 1k 20.4065 2 13 2q
10.6140 3 2 3d 20.5379 1 18 lw
10.9042 3 0 3s 20.5596 4 7 4j
11.0703 2 5 2h 20.7960 3 10 3m
11. 0791 1 9 11 21.2312 5 5 5h
11.9729 3 3 3f 21.5372 2 14 2r
12.1428 1 10 im 21.5779 1 19 ix
12.2794 2 6 2i 21.6667 6 6f
12.4046 4 1 4p 21.8401 4 8 4k
13.2024 1 11 in 21.8997 7 1 7p
13.2956 3 4 3g 22.0000 3 11 3n
13.4721 2 7 2j 22.5781 5 6 5i
13.8463 4 2 4d 22.6165 1 20 ly
14.0663 4 0 4s 22.6625 2 15 2t
14.2580 1 12 lo 23.0829 6 4 6g
14.5906 3 5 3h 23.167 4 9 41
14.6513 2 8 2k 23.1950 3 12 3o
15.2446 4 3 4f 23.3906 7 2 7d
15.3108 1 13 lq 23.5194 7 0 7s
15.5793 5 1 5p 23.6534 1 21 Iz
15.8193 2 9 21 23.7832 2 16 2u
15.8633 3 6 3i 23.9069 5 7 5j
16.3604 1 14 ir 24.3608 4 10 4m
16.6094 4 4 4g 243824 3 13 3q
16.9776 2 10 2m 24.4749 6 5 6h
17.0431 5 2 5d 24.6899 1 22 la
1 "1 1 "7 "i "7 ", 0 7












Acoustic


Spectrum Analysis


acoustic


resonance


spectrum


will


typically


show


series


freauencies


that


represent


resonances


allowed


eqn.


3-21.


However,


acoustic


spectrum


may


also


contain


other


frequencies


which


are


extraneous


noise.


Consequently,


necessary


employ


stratagem


distinguishing


propagating medium


acoustic


from


spectrum


other


sound


features.


Assume


that


given moment


isolated


system,


speed


sound


a constant.


Then,


eqn.


taken


different


eigenvalues


yields


2xa


v1


Eqn.


2xa


3-22


From


equation


3-22


, it


can


seen


that


ratio


V9 ,m/5i, m


should


constant


resonances


which


involve


only


vibration


contained


fluid.


A typical


acoustic


spectrum


shown


figure


3-1.


particular


peak


thought


fluid


resonance


frequency,


division


correct


eigenvalue


followed


multiplication by subsequent eigenvalues should provide values


Cl .l .


--~A n


Cuhrr~hnn a


- ~ I I as PI-~~ -,nr r n, rt r F Hil' rJ T 4Ir I I -af II i I I


- -2l


. q


I













fluid normal


mode.


such


fashion,


acoustic


spectrum


fluid medium


can


always


'singled


out'


from the


background


noise.


Figure
cavity


Acoustic


three


inch


spectrum


methane


from


spherical


radius)


Figure


shows


typical


resonance


peaks


more


detail.


Peaks


correspond


, 2d


r a a r n fl a


racflflr. Ira -I1


'72 Ilioc


r~tin


via.


I %I f rv N rV" .f L.J irv 1 Iu II LUt rv .. VL t L. At 2 1.U .. .I.. *L.L *% .


3.5E+006 -
-4
-4
I
3 OE+006 -

3.5E+006 --

.E+006 -
-e
3 2.OE+006 ,

F-

I
S 1.5E+006


l.OE+006 -

1
5.0E+005 -


O.OE+0O00 H-------- UI
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Frequency, Hz













2E*005


1E-005i


4E+0051


4E+0051
1


1 -005
-J
E
S


5E-004


2E-+-OO&


2E+005


OE000---
2447


2547


OE 0 -~OOO


2577


2677


OE+0001-
3174


3274


OEc00-i-O --
47712


Figure 3
frequency i


Typical
(excerpts


detail


individual


resonance


from


perturb


observed


frequencies.


Mehl


Moldover23',4


have


considered


most


prominent


these


and suggest


that


equation


should be


expanded


following manner


2xa


+ Avs
s


+ Avel


+ALv,


+ Vgeom


added


terms


represent


perturbations


thermal


boundary


layer


effect


resonator


wall,


finite


elastic compliance


resonator wall,


the effects of


filling


tubulations


connected


resonator


wall


and


deviations


from


perfect


sphericity


resonator


wall.


carefully


designed


fabricated


apparatus,


these


perturbations


can


quite


small.


Corrections


can


made


either


theoretical


treatments


or by


calibration.


-- -- -- a


- CCh n


A4 'F.Fr~rrnn


mrn rl A


TT


rIo rlar II' f1-d ,-. Ir 111 Ir" I rrIIIV~izn


vibration


4872


I II I


s











modes


involve


tangential


motion


fluid


resonator walls and are not subject to the effects of viscous


drag.


Such resonances


are


generally very


well


defined and


resolved from other allowed resonances.


This facilitates easy


identification and subsequent tracking.
















CHAPTER


EXPERIMENTAL



Procedure


A spherical


acoustic


resonator


assembly


with


a variable


volume was utilized


as a probe to


locate the critical point of


methane.


Temperature


pressure


sonic


speed


were


primary


variables


that


were


tracked,


volume


was


also


followed


non


analytical


manner.


None


variables


was measured directly,


but each was followed as a more readily


measurable


related


parameter.


Figures


are


schematic


representations


represents


physical


overall


apparatus


experiment.


while


Figure


figure


represents the control


instrumentation.


later reference,


the tubulations,


are


referred


pump,


resonator,


sealed methane


inner bellows and RTD shroud


'system


Several


Sections


apparatus


have


unique


features


that


must


addressed


separately


highlight


construction


detail.


Calibrations


temperature


pressure


measuring


devices are


included by necessity.


Also,


instrumentation and


accuracy


are


documented


illustrate


level































































- e-


-


;t

-A



















































rn

77


,nK


'I'


___


i












Initial


Loading


A brief


overview


experimental


procedure


allows


working


insight


into


technique.


Research


grade


methane


assayed


99.99%


pure


from


Scott


Specialty


Gas,


Inc.


was


utilized


experimental


fluid.


entire


system


contact


with


methane


was


stainless


steel


construction


minimize


the potential


for chemical


reaction with containment


walls.


To ensure


initial methane


purity within the


system,


regimen


system


purges


was


conducted


following


manner...


valves


were


closed


except


bellows


bypass


valve,


system


shut


valve


the methane


shut


valve.


The mechanical


shut


vacuum pump


valve


was


started.


mechanical


vacuum


pump


was


opened.


entire


system


was


evacuated


period


of several


hours to ensure the system pressure


fell


below


10-3


torr.


shut


valve


the


mechanical


vacuum


pump


was


closed.


leak


valve


methane


tank


was


opened


ymn nut ci


Snt- rn ri


c1nw hr


I \ I I


mPth;lnP


I / 1











leak valve to the methane tank was closed.


The system pump was started and allowed to run


about


twenty minutes.


entire


procedure


was


repeated starting


step


three


a total


iterations.


The charging procedure required a period of approximately


thirty


hours.


bellows


bypass


valve


was


then


closed


duration


experiment


isolating


one


side


bellows


from the other.


liquid nitrogen tank was attached


constant


temperature


environment


and


system


was


cooled.


The constant temperature environment RTD,


(Resistance


Temperature


Device),


was


monitored


until


system


temperature


fell


near


temperature


anticipated


was


then kept


critical


constant


value.


within


Environment


millikelvin


through


use


a computer


control


algorithm.


methane


leak


valve


shut


valve


were


reopened


minutely


bring the


system pressure


to a value


close


possible to the anticipated critical pressure


The argon leak


valve


shut


valve


were


similarly


opened


maintain


comparable


pressure


both


sides


fragile


bellows.


prevent


rupture


bellows


excess


pressure,


bellows


position


was


constantly


monitored


through


the use of an LVDT


, (Linear Variable Differential


Transformer,











Ap70psi.


order


safeguard


system


the event


dangerous


pressure


gradient,


copper


rupture


disc


lower


strength


was


utililized.


Once


system


pressure


was


near


anticipated


critical


value,


valves


except


system shut


valve,


the argon


shut


valve


argon


leak


valve


were


closed.


Argon


gas continued


to be introduced to the back side of the bellows


order


minimum posit


compress


ion,


slowly.


the argon shut


reaching


bellows


valve and the argon


leak


valve


were


closed


argon


bleed


valve


was


opened


atmosphere.


system pump was put


into operation


to assure


system


homogeneity.


Sufficient


time


was


allowed


temperature


equilibrium


established.


argon


leak


valve


was


reopened


allow


a slow


bleed


of bellows


back


pressure.


The movement


of the bellows from minimum to maximum


position provided a variable volume capability.


Sweep time of


the bellows


full


range of


motion was typically


slow


ten to


twelve hours to ensure only small departures from equilibrium.


During the volume sweep,


the temperature was followed with the


system RTD,


pressure


was


followed


with


Heise


digital


Bourdon gauge and the acoustic


spectrum was


followed with


sonic transducers.


All three instruments were serially polled


results


recorded


periodically


one


two


minute












Attainment


System


Critical


Density


Unfortunately,


a proper


system


loading


methane


could not be obtained simply by filling and cooling the system


to the anti


cipated critical pressure and temperature and then


sealing


off.


Even


small


error


attempting


match


system pressure and temperature to critical


values was greatly


magnified


out


proportion


pressure


change


approaching


zero


critical


point.


purpose


this


rese


arch was to use sonic


speed


as a sensitive detector of the


critical


state.


therm


dynamic


speed


sound approaches


zero


rapidly


temperature


the critical

density, (or


point

molar


where


volume),


the

are


pressure,


their


critical


values,


system density


(see


were


chapter


below


two)

critical


Subsequently,


value,


each new value


system


speed


sound


would


increase


bellows


was


swept


outwards.


Conversely


system


density


were


above


critical


density,


each


new


value


system


speed


sound


would


decrease


bellows


was


swept


outwards.


Only


when


system


density matched


critical


density


somewhere


middle


the bellows


sweep did


speed


sound


first


decrease,


reach


a minimum


and


finally


increase.


,*~~~~~~'t 0 I-r 7 0 nlI I


cnnc&


c~lniir


c~tct nm


I I J


nrnnh~


r i i


'I' nn


~ rnr


r- n i _











bleed


above


argon


tank


was


opened


atmosphere and the argon leak valve was opened sufficiently to


allow


course


bellows


ten


sweep


twelve


from


hours.


high


Temperature,


volume


pressure


over


and an


acoustic


spectrum


were


taken


about


every


two


minutes.


Only


about t

whether


;hirty

system


:otal


data


density


points


was


high


were


required


low,


data


determine


acquisition


was


halted


after


about


hour.


bleed


valve


above


argon


tank


was


then


closed


argon


tank


shut


valve


was


opened.


Argon


was


fed to


the back side of


the bellows


order to


recompress


that


point


argon


valves were


shut


system


was


allowed


sufficient


time


resume


constant


temperature.


mean


time,


raw


data


were


transferred


data


manipulation


computer


where


acoustic


from


signals


time


could


domain


undergo


into


fast


Fourier


frequency


transformation


domain,


(i.e.,


acoustic resonance spectrum)


Each acoustic


spectrum was then


viewed


and


resonance


peaks


identified.


Individual


peaks


were


then


tracked


from


one


spectrum


next.


Once


system


density


.rend i

was


the speed of


adjusted


sound was


accordingly


identified,


either


adding


methane


bleeding


methane


from


system.


entire


process


was


then


repeated


until


elusive


minimum












Data Acquisition


actual


acquisition


data


after


loading


system


proper


procedures


density


required


was


largely


data


uneventful.


acquisition


had


already


the

been


enabled.


Initially,


bellows


began


stationary


minimum


position,


master


data


computer


was


brought


monitoring state,


and sufficient


time was allowed for thermal


equilibrium


established


throughout


the


system.


argon bleed valve


was vented to atmosphere


and the argon


leak


valve


was


adjusted


achieve


total


bellows


sweep


time


ten


twelve


hours.


When


initial


bellows


movement


passed


certain


preset


value


just


above


its


starting


point,


monitoring


master


dat a


computer


switched


active


dat a


acquisition


state.


The master


data


computer


triggered


function


generator to


produce

over a


sine


period


wave


with


about


six


frequency

seconds.


sweep

The fi


from


Junction


10kHz


generator


also

the


produced


Analog


synchronized


Digital


Converter,


square


wave


(ADC),


that


and


triggered

Slock-in


amplifier to begin


data acquisition.


swept


sine wave was


utilized as an acoustic excitation source


for the


input sonic


transducer.


acoustic


signal


was


transmitted


from


one


- .. I


C' l


S- I 4- -


I-- l 'S -r~r.A ~*1- 1t' n mntn n rt nn nr flr TTfi fC I iiIII'.l


Cu~nn rj~rnnu


I-- A A -


nn *r~h









39

pass filter which allowed only the transmitted frequency that

matched the excitation frequency to pass on to the ADC. The


acoustic spectrum was converted from an analog to a digital

signal by the ADC and recorded by the master data computer.


Immediately


afterward,


DMM


(Digital


Multi


Meter)


that


tracked the system RTD,


the DMM that tracked the LVDT


, and the


ADC that tracked the


Heise gauge were serially polled and the


results recorded by the master data computer.


The master data


computer then returned to its monitoring state and awaited the


bellows


movement


pass


next


preset


position


that


triggered the next iteration of the acquisition procedure.


this


way,


data


sets


were


acquired


about


every


minutes until the bellows movement reached a final value just


below


bellows


maximum


position.


dat a


acquisition


program then self terminated and sounded an alarm to alert the


operator that the bellows movement needed to be arrested.


argon bleed


valve


was


closed


and


argon


shut


valve


reopened


long


enough


recompress


the bellows


slowly


to a


minimum


adjusted


position.


achieve


temperature


different


control


working


program


temperature


was


and


sufficient


time


was


allowed


thermal


equilibrium


reestablish.


The entire procedure was reiterated at different


temperatures to span the critical point parameters.












seventeen bit


fast Fourier transform was performed on each raw


acoustic


signal.


The acoustic data


were transformed from the


time


domain


frequency


domain.


Each


spect rum


was


individually


examined


identify


acoustic


resonance


peaks.


individual


peak


frequencies


were


then


tracked


from


one


spectrum


next


order


follow


system


speed


sound.


A data bank was constructed which contained the system


speed


sound,


system


pressure,


system


temperature


system


volume.


Finally,


entire


data


bank


was


saved


on tape


storage


media


future


reference.


table


4-1 at


the end of this chapter for an instrumentation summary.


Calibration


Heise


Pressure


Gauge


Calibration


order


locate


critical


pressure,


was


essential


utilize


accurate


pressure


gauge.


Heise


pressure gauge that was used in this experiment was calibrated


just


before


immediately


after


data


sets


were


acquired.


Since


was


located


in proximity


cooled,


constant


temperature


environment,


final


calibration


was


performed


with


constant


temperature


environment


same


temperature


which


experiment


was


run.


fl ^ n v 1 n


IA7~C I I iiti i~i 7fCT C


nprfn rmmr


d~aci


wei aht


TA? l C


/^


I )









41

was floated atop the piston and provided an absolute pressure,


(p=mg/area,


to gravity)


where m


, on the


the mass and g


fluid within


the acceleration due


the calibration system.


weights


were


loaded


under


vacuum,


correction


pressure


air


was


necessary.


Heise


gauge


output


was


analog


voltage


which


was


converted


digital


format


that


could


acquired

weight p


pressure


master

gauge


data

was c


computer.


compared


The

the


mass

ADC o


dead


>utput


equation was generated that

an absolute pressure measure]


converted ADC units directly


The actual


into


calibration plot


shown


figure


4-3.


The calibration plot for the Heise gauge was incapable of


being


extended


pressure


ranges


involved


this


experiment.


Each


weights


used


dead


weight


pressure


gauge


was


certified


manufacturer,


calibration range extended only to


psia while experimental


pressure


ranges


extended


approximately


fifty


higher


However


extreme


linearity


calibration


line


(chi


squared


lends


a linear


confidence


regression


to extrapolation


data


over this


was


small


.99999952),


range.


The e

calibration


experimental


arrangement


illustrated


figure


used

4-4.


for

The


the

high


pressure

pressure










42


The values given by the ADC and the masses/absolute pressures


dead


weight


pressure


gauge


were


used


generate


figure


4-3.


- ~Sii r


roti


,eI'S


-r
x/ '


2000


-~ r r0 r


* fC'C


1000


I i i i i I II I i1 [ t 1 I .. I I I I I


400


600


Pres


sure,


p510.


Figure 4-3.


Calibration


plot


Heise


gauge


vs.


units.





System RTD Temperature Calibration


RTD'


were


utilized


experiment.


isothermal environment RTD was used as a feedback device for


thc t-mnrrnaitir rnnt-rnl cnmnut pr


The svstem RTD was used to


T












The system RTD


recei


ved the closest scrutiny and underwent the


most


stringent


calibration


process.


Figure


4-4.


Pressure


calibration


setup.


A single platinum RTD was purchased precalibrated by


manufacturer


complied


with


guidelines


set


International


Temperature


Scale


1990.


25 It


came


complete


with a,


p and 6 coefficients


for substitution


into a standard


rna1-Tinn r atl-ina the


temoprature


to measured RTD values.


Constant
Temper at ur e
Env Ironment
Argon Tank Envronen


Ruska
Dead We eight N
Pressure Gaude I


To Vaccuum


Rus ka Pu.p


Heise Gauge


- -












:em


-Te


- -T
9 C~I I


v-qt


te(i


NB


-
~


System


: vibratedd RTD


' 7T1Iii1iT7 TiiT1I:iiIiTiI rl iiTIII Fij ii T '


:o00


Time,


1200


secs.


Figure
under


4-5.


Response


identical


system RTD


precalibrated RTD


condition


ensure


close


temperature


experimental


calibration


conditions


conditions


possible,


were


both


environment


system


were


placed


alongside


precalibrated


constant


temperature


environment.


temperature


control


computer


was


used


drop


temperature to the range of this experiment.

the response of both the system RTD and the


Figure 4-5 shows


precalibrated RTD












precalibrated


RTD.


slight


departure


from


linear


dependence


is easi


ly accounted for by a simple polynomial fit.


S t em


KT TD
KTC:


-- -4
(Vtj~ aaI6L


ca ibrated


TD.


0 80
o 8o


110


i I T120
120


recal


ibrated


RTD


Resistan


Ohms.


Figure 4-6


Plot


System


vs.


Precalibrated


exhibiting strong linear response at low temperatures.


precalibrated RTD


was


electrically


connected


Keithley


196A DMM.


As a check of the DMM,


the resistance of


the precalibrated RTD was measured within a water triple point

cell to ensure a resistance value that corresponded to a true


temperature


value.


triple


point


water


been


established


as equal to


73.1600K.


!2C2
-
-;
-4
r~-4


I









46


Any difference between this lowest resistance value and the

manufacturer's precalibrated value for the same temperature


was treated as an offset due to DMM miscalibration.


offset amounted to a subtraction of 0.3529K from each recorded


temperature


value


was


accounted


data


tabulations.


~~t -l
~1 S -r


'.3T


21 1 ~~i C G r:


212
I.- '-


("C)



?1 99C5C





99.9~C



- t.- ., V


~1
-1
-4


-4
t
2
H
-4


Triple point of water
= 273.1600K
- 99.8460ohms 0 022chms


V I F TV


"note


400.00 600.00 800.00 1000.00 1200.00 1400.00


Time,


secs.


Figure 4-7.


Triple


point


water


calibration


for the


precalibrated RTD.


SOiC~


7


: 1 i 1 i 1 1 ? i i I i 1 1 1 'T1 I I TT r T ~n? 7~rr ij j 1 I i 1 i i TT i i i 5 i T i i i i i I I











Apparatus


Detail


Resonator


Figure


4-8.


Spherical


resonator


schematic.


heart


apparatus,


spherical


resonator was


machined


with


highest


possible


accuracy


available.


inch


diameter


resonator


had


total


diametral


runout


.0001


inches.


construction


was


type


stainless


+t- a -1 -hynrriinhnit-


- nirm 4-Rt


i a sc hematic of the resonator.


K~j~


(___________Lu











being


system


refilled,


pressure


apparatus


climbed


was


high


allowed


eighty


warm


atmospheres


prevent


experimental


apparatus


from


behaving


bomb,


upper


lower


collars


shown


figure


were


clamped


together with


twenty


-24 bolts each torqued to one hundred


foot


pounds.


thin piece of


material


in the center


of the


drawing represents a


copper gasket


that


sealed the two


halves


chamber


together


being


swaged


into


a groove


in the


upper


lower


hemisphere.


shown


figure


are


small


holes,


one


top of the chamber and one in the bottom of the chamber


which


allowed fluids


to enter


experiment


was


and exit


pumped


the chamber.


continuously


fluid used


throughout


system to ensure good blending and prevent


the development


temperature


gradients


actual


system


pump


details


have


been


documented


elsewhere27


pump


circulated


system


fluid


without


generating


strong pressure


gradients.


Transducers


transducers


themselves


were PZT bimorph


oscillators


.400


inches


diameter


.050


inches


thick.


Oscillation


frequency


electric


variation


potential


were


across


the


induced


excitation


placing


transduce


variable

r. The











sender,


does not


interfere with


the spherically symmetric


radial modes,


which are often the most favored signals.


Since


radial


modes


vibration


are


affected by


tangential


interaction


with


chamber


walls,


they


are


often


most


attractive for scientific scrutiny.


A transducer placement of


ninety degrees gave the highest probability of


clear reception


of the most


unperturbed modes of vibration


within the sphere.


Proper


mounting


transducers


within


chamber


required


careful


attention.


electric


signal


applied


transducer.


electric


had to be


received


stainless


potential


electrically


from


steel


return.


shielded


least


chamber


one


itself


electric


from


rest


side


could act


potential


input


chamber


and had to be mounted so as to withstand a pressure as high as


100 atmospheres.


Also,


the electrical


and mechanical


contact


with


transducer


had


have


dampening


effect


else


vibrational


signal


strength


would be


poor.


A machinable


ceramic,


MACOR,


was


used


fabricate


part


of the transducer assembly.


This material


was elected for its


structural


strength,


fluid


impermeability


and


electrical


insulating


quality.


tapered


ram


plug


was


prepared


from


MACOR


match


tapered


seat


machined


into


spherical


chamber


wall.


A sheath


of malleable


copper with


an integral













into a


tapered hole


formed coaxially


in the ceramic MACOR


plug.


Copper




MACOR


*1


Sheat h




Ceram c


s St.


Pin


With


~'wa


sweat e d


gea
ath


ble Copper

Seal


Copper


Sheath


PZT


Electr


Osc i 1 1 ator


. Connect


Open


allow

Sound


sage


Tens ion


Chamber


CrSr.- m c


Gr o und


Figure


4-9.


Construction


detail


transducer mount.


metal


sheath


MACOR


plug


and


centerline


pin


were


assembled


applied


and pressed


pressure


into


greater


the matching


than


tapered seat


maximum


with


pressure


generated


experiment.


action


pressing


*, ,-~~ Y- C' r r~ ~ nfln fl 1 c fY2C CC i-IIk


+- ,hnara


f.i nP


mpl-al


~
/ ~J7


cw~np~


*


1 V T


I nnn mk itr


I 6--


I 1 LJ


h fnn











Once an electrically


isolated


contact


was provided,


each


transducer


was


supported


bottom


electrical


contact


its center.


Center transducer support


was a novel


mounting


technique


previous


workers


had


used


more


common


edge


mounting


technique.


Center mounting


allowed


transducer


float


unconstrained


except


point


contact.


A vibrating circular membrane has a vibrational node


directly


center.


Therefore,


transducer


contact


with


electrical


connectors


center


transducer


caused very


little dampening effect.


This arrangement proved


give


better


performance


terms


enhanced


sensitivity


and prolonged


oscillator


life.


System RTD


Placement


system


RTD


was


placed


within


shroud


that


was


direct


contact


with


system


fluid.


system


fluid


was


continuously moved through the shroud to ensure disruption of


any potential


temperature gradients.


The shroud wall was made


thin


system


pressures


would


allow


provide


rapid


thermal


equilibration across the shroud barrier.


Construction


detail


illustrated


figure


4-10.


Bellows


Construction


Detail























Figure
shroud


4-10.


Construction


detail


system


temperature


experiment.


The bellows


itself,


however


was


fabricated from


very


thin


material


allow


flexibility.


keep


relatively fragile bellows intact at extreme pressures,


fluids


were


introduced to both sides


of the bellows at approximately


equal pressures.


System pressure was


isolated on


one


side of


the bellows while argon balancing pressure was isolated on the


other


side


bellows.


thin material


the bellows


walls


was


sufficient


to contain


fluids


on opposite


sides


provided


pressure


difference


never


exceeded seventy


psi.


Approximately


a thirty


pressure


difference


from


one


side


of the bellows to the other was enough to compress the bellows


to its minimum position.


System volume variation was achieved


slowly


redu


cing


argon


balancing


pressure


back


side


bellows.


system


pressure


was


then


able


push


back


bellows


achieve


increased


volume.


Temper at ure
Shroud


RTD


SyC ill


j---
V;















TV1


illll~lLllllbhll~lhhmliT~~i

Ki
H-


I I


~-TT-=Ii


i


IIYYIYY''Y'1 5 __________irrr


C-:


L L2


I


~ "


r


-











54



Cd



V c),E 42 ()- .4 4 (1) 0 d
c~ka C Wr C WC 0' -H )k )k I
C r r o C I) 0) 00) 4~l I II) ~
EC4E Sd SC Sb CCS Cebl 0Ir ,





cOtW> OUWe-d)-d



>1 4 0) 4,a a

C r i-H CC W~O CA rH' 4 (i
fl r-Hr-4,-c aAt S, U r6r 0
U W c ccra -Hd CO4 dJ WC4t
U~~a O3W' 00) -HCi 0'WliC S







rHW.CO C wCC: 0




da aW-H-flE W-H02 C>UE 4-J.QE ,
WWc (dO OJW~rd0 WIk rHOb l O
k~SLR k$SO~ I ,klr O


rl II II I fl


o LC) '.0 LC >IIr cI
rH n-i r I Cd




7 rHC rHC r-H g



c, -H -H4 -H cv



4d C
C, dc (0c I-
0), U)( 0) 0) 0LHI











55






U) S-r CoQ

C, C W0 COG) 0"' 0) c
4JH4J-cl d-r (dO G)OU HHdCOrb

U) 4( i U) -CI I -H Ck C) tmC01 Ir Q.1 U)

C, 0I U~ >, i- > C, Ct Q. 0"' Cd~k a, r-4 cu $.If-
WouXOH3eC(d 53inr HCd-Wr..JWfO4-






o" -Ht oi

O U d -I U
c i) r- 04(/)3 c
o 0r1W7EdW 4




+Id>4- +104-4 H



'T' O)NN-Hr-1Q
>)3 ZI3:rH0 Q


a, ~rH(d O
pr;)~ c X3
0 Co '0 tr noO
*HSkO40CW( W
f-44-3(00Q4 Or-Ior


H-
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Ti to H
o NI tOI r
>2 ('1
(' U)C

4-3 tr




U'c rr
COU


Cd 0) 4WW0 r
>2 (0 >c C


5 0
P) U
C
(1) nN O 5- 4kI











56






0)- >1 -Ha
UI) 4-) C COO r
a) -ric HC 0~ rH 4J c


40(RC)-HaCo ( '04 -O C) 4Jr

Oa trd C4-' COCCI) COc)
CWCUWXO HXWx~d






C4 rHOJ
0i -rlrHl
SC C d U)I Ir
U U N
U '0-H HQI ) WrH
Q~~O WC0 0>
rH~ >(drj Z'0

C)o w ~ 1




U>-HN C N0E

CC) Nc, ?Z;O rWI ,

Cr(1 -Hocd 3 a


S $- ct -ri co o- Q.





o to N4c


s-Im



C)



0H iE s- --

C C~f- -ci C)
4-c)> WC C d Cr
C

r-I 0

CUr
















CHAPTER
RESULTS


Spherical


Resonator


Spherical


resonator


development


study


fluid


speed


sound


related


properties


been


successful


activity


workers


this


research


group.


29-38


There


were,


however,


limitations


previous


work


primarily


construction


spherical


spherical


resonator


resonators.


construction was of


earliest


aluminum and designed


operate


primarily


near


ambient


temperature


pressure.


Later,


stainless steel


was used as a construction material


order


study


natural


mixtures


which


contained


small


amounts

designed


corrosive gases.


operate


These


moderate


spherical


temperatures


resonators


and


were


pressures.


They


suffered


from


imperfect


spherical


geometry


fabrication methods


used by the


outside manufacturers.


current


apparatus


found on earlier


incorporates


instruments.


Newly


several


added are


features


thick wall


design


high


pressure


work


exacting


tolerances


ensure spherical


geometry


Operating temperatures can be from


- ~ w- -% 4. -% -r T -f .- %.


-,, -.L


C~Ctr


~r uA~nIYlnn


J


II L


nc~n











stainless


steel


construction


used


throughout


closed


resonator


system.


In addition


to the extremes


in operating


conditions,


apparatus


also


incorporates


a unique


feature


that


allows


system


volume


varied.


expandable


bellows


assembly


that varies the system volume up to 20


is an integral


part


the apparatus.


The addition of the bellows for the first time


make


s isothermal


or isobaric studies practical.


If isochoric


work


system


operating


desired,

with c


bellows


constant


pressures


and


can


volume


easily


still


temperatures.


bypassed


capable


For


this


to yield

extreme


report,


work


was


done


under


controlled


isothermal


conditions.


Acoustic


Spectra


typical


complete


data


acquired


isothermal


volume


scan


included


approximately


80-150


data


points.


Each


datum


point


consisted


a value


temperature,


pressure,


volume


plus


entire


acoustic


spectrum.


acoustic


spectrum consisted of a plot of signal amplitude versus signal


frequency.


Range


and


resolution


for these


variable


was


0-65535

Figure


units


unit


contains


a typical


-10000Hz

acoustic


+ 0.15Hz


respectively.


spectrum taken


from a


data


collected


well


away


from


critical


point.














4.5E+ 006

4E+006


3.5E+ 006

3E+006

2.5E + 006

2E+006

1.5E + 006


1E+006

500000

0


2500


5000 7500


10000


Frequency, Hz .


Figure


5-1.


critical


Typical acoustic spectrum recorded well away from
point.


Each

resonant


spectrum

vibrations


was


visually


method


checked

outlined


identify


chapter


three.


Once


identified,


third


radial


mode


vibration


was


tracked


from


spectrum


spectrum


follow


changing


system speed


Figure


sound.


is an acoustic spectrum recorded very near the


critical


point.


Notice


significant


reduction


signal


compared


figure


5-1.


most


prominent


signals











been


dampened


dispersion


proximity


critical


depressed


point


sonic


Further


speed.


analysis


Both


consistent


reduction


with


resonance


amplitude


and


sonic


speed


are


consistent


with


anticipated


behavior


near


critical


state.


Figure 5-2. Typical acoustic spectrum recorded near the


critical


point


Figure 5-3 and figure 5-4


are excerpts


from two acoustic























ii ii
- -*AA"- --


* -


- --e

Vb


4000


4500


5060


Frequency, Hz.

Figure 5-3. Excerpt from an acoustic spectrum near the
critical point.















'I "







*=.- -
?c f ;- .........











second spectrum shows


shift


toward zero


frequency


well


an overall decrease in signal amplitude.


The frequency of vibration for a resonant peak is related


system


speed


sound


according


3-21.


addition,


the thermodynamic speed of sound becomes zero at the


critical


point.


Therefore


acoustic


spectrum


with


lowest


frequency


vibration


given


resonant


peak


corresponds


experimental


conditions


closest


critical point for that data set.


In figure


and figure


4, the peaks at 4612Hz and at 4517Hz respectively are each due


third radial mode of


vibration.


information


both spectra was recorded just two minutes apart at a change


system


volume


only


0.1%.


Clearly,


shift


third radial


mode


vibration


frequency


both


dramatic and


quantifiable


Figure 5-5


is a superposition


of both spectra excerpts


illustrates


not


only


frequency


shift


third


radial


mode


also


loss


signal


amplitude


that


accompanies the approach of the critical point .


The placement


chamber


transducers


was


designed


maximize


ability to identify the radial modes of vibration.


Since the


radial modes of vibration correspond to symmetric passage of


sound toward and away


from the chamber walls,


they


are




































Figure 5-5. Superposition of two acoustic spectra excerpts


illustrating


amplitude
critical


that


rapid


accompanies


change


a small


resonant


change


frequency


volume


near the


point


third


radial


mode


vibration


was


utilized


throughout


the experiment


for calculating the


system speed of


sound for two


reasons.


First,


readily


identifiable and


well

large


resolved

height


from


neighboring


width


ratio,


signals.


value)


Second,

which


makes


relatively


easy


track


from


one


acoustic


spectrum


next.


Data


Manipulation












frequency


vibration


was


converted


system


speed


sound,


(see


eqn.


3-21),


combined


with


other


system


variables.


plot


system


speed


sound


versus


system


volume


illustrates


relationship


that


exists


between


two


variables.


Plot


versus


system


speed


approximate


sys temu


sound


volumee.


Pt-t


Q1OAIO


2(10 Ut


190.00


jIUt. Ut


170.00


1BU.UO


Approximate


Volume,


Figure
sound


5-6
and


Typical


system


volume


relationship


near


between


critical


system


speed


temperature.


allowed


modes


vibration


shift


their


frequencies


strongly


critical


downward


density.


While at


system


density


or near the critical


approaches


temperature,


the v.c;stm seed of sound exhibits a similar behavior as


it is













system


speed


sound


system


volume


under


these


conditions.


Plot


versuS


system speed
proportional s


sound


;ystemi


density


A-u ~


l 91


170 U

11-A
160 -


/Volume,
1/Volume,


(Proportional


.density)
density)


Figure 5-
sound and


Typical
; system


relationship between the system speed of
density.


closed


system,


molar


density


inversely


related


system


volume


moles/V.


Furthermore,


molar


volume


defined


volume


occupied by


a mole,


= V/moles.


Therefore,


the molar volume and molar density


are


inversely


graphically


related,


terms


1/p,


density


data


figure


shown


5-7.


C --.


.r-~r~ ni"1


In nn 'T1 r I 'tll'l r f ll Ir 11 Afl In lf & l I VtJII tl~lc.uL A.


a1rnr


U.31T


0.40


1-4-i


r7]ntta~











the amount


of methane introduced to the system was devised and


fabricated.


calibration


scheme


metering


exact


system


volume


a function


bellows


position


was


also


devi


and


into


practice.


Unfortunately,


feature


the bellows that


responsible


the experiment


defeating


was built


ability


to exploit


measure


was also


system


density


accurately.


walls


bellows


were


quite


thin


flexible.


such,


they


thereby


bellows


allowed the

change the


was


bellows

overall


position,


move


system

walls


freely

volume.


were


When


compressed


down

the

and


quite


rigid.


However,


when


bellows


was


high


position,


the walls


were uncompressed and had some freedom of


movement


temperature


expand


was


changed,


bulge


outwards.


stiffness


system


compliance


bellows


walls


also


changed.


Therefore,


amount


bellows outward bulge and the change


in system volume changed


dependent


temperature.


extra


degree


freedom


bulge outward combined with changing wall

temperature made it impossible to find an


properties based on


exact system density


by monitoring


bellows


position.


Consequently,


while


volume measurements illustrated in various graphs in this work


are


highly


accurate,


they


have


uncertainty











system


volume


was


precisely


quantifiable,


desirable


focus


system


parameter


that


was


more


accurately measured.


constant temperature environment was


held


within


ten millikelvin


each


set


temperature


under


study.


system


volume


was


increased


a nearly


constant


rate


and


system


pressure


and


system


speed


sound


were


tracked


analytically.


Figure 5-8. Typical relationship between the system speed of


sound and


system pressure.











Figure


plot


system


speed


sound


versus


system


pressure.


minimum


that


appears


center


the graph corresponds to


the closest


approach


to the critical


point recorded by the system parameters;


that


is the pressure,


volume,


temperature and the speed of sound within the enclosed


methane.


Figure


may be roughly divided into two parts.


To the


left


minimum,


system


pressure


less


than


critical

phase fl


pressure


uid.


and

this


system


region,


exists


system


single


speed


gaseous

sound is


inversely


proportional


system


pressure.


pressure


increased,


speed


sound


decreased.


the right


the minimum,


the system pressure


is greater than


critical


fluid


pressure


liquid


system


equilibrium.


exists


this


phase


region,


system


sonic


speed


directly


proportional


system


pressure.


As the pressure


is increased,


the speed of


sound is


also


increased.


Each plot


of system speed of sound versus system pressure


can


examined


location


minimum.


ince


thermodynamic


speed


sound


goes


zero


critical


point,


individual


plot


with


minimum


that


extends


lowest


system


speed


sound


corresponds











plot


that


most


closely


approaches


critical


point


conditions.


At the same time,


an indication of the precision


of the overall technique can be demonstrated.


Plot


all


data


runs


on


common


set


axes.


n r~ -'


190 00

180.00

170.00


S1r60 00


T T I -i i i T I I


;~~ I 7jT


4 5500


4 6000


a 6500


4.7000


4.7500


Pressure


, MPa.


Figure 5-9.
sound on a


Plot of


common set of


data sets of pressure and speed of


axes.


Figure 5-9


illustrates


the asymptotic nature of


system


speed


sound


experimental


conditions


approach


critical


point


conditions.


Combined


plot


shown


- I


-_i











exhibits


'spike'


pointing


pressure


that


corresponds


critical


pressure.


While


true


that


system


speed


sound


never


reaches


zero,


the inaccuracy of identifying the critical point


is very small.


The asymptotic behavior exhibited by the speed


of sound at


critical


the critical


pressure


point


that


provides a


precludes


strong


indicator of


necessity


attain


zero


speed


sound-


data


this


experiment


were


to be collected


under


isothermal


conditions.


constant


temperature


environment


was


kept


at a set


point


temperature


to within


ten millikelvin


for each data


run.


However,


temperature within


the fluid


system itself


did not


remain nearly


so constant.


Even


though


it was possible to maintain a very constant temperature around


sealed


spherical


resonator,


fluid


inside


system


was


isolated enough


that


temperature variation was as much as


several


hundred millikelvin


over the


course


of a single


data


run.


Figure 5-10 illustrates the temperature variation within


the sealed spherical


resonator


as a function of


time during a


data


collection


run.


apparent


from figure 5-10 that


it was difficult


maintain


isothermal


conditions


during


a data


collection


run.


Again,


one


of the


strengths


of the


system was


also one of














Typical


over


the


variation
course


temperature


entire


data


run.


190.000


r180.750
Q


189.500


*89.250


0.00


100.00


- i--- I


200.00


300.00


400.00


500.00


800,00


Time,


mins.


Figure


5-10.


Largest


temperature variation for


a single data


run.


work


against


surroundings.


process


essentially


adiabatic


since


exchange


heat


across


phase

slow.


boundary


lC


represented by

)ss of internal


solid


l


energy


system

that a<


walls


Companies


very

the


expansion


thermodynamic


fluid


reflected


overall


such


drop


a phenomena


system


temperature.


would be


to provide


only


sufficient


to prevent


amount


I II


I i i i


mi











conditions.


Consequently,


experiments were carried out


under


approximate


isothermal


conditions


only.


more


efficient heat exchange system would have to be provided for

future work to approximate isothermal working conditions more

closely.


Such


situation


does


negate


results


experiment,


because


system


temperature


was


recorded


throughout


data


collection.


plot


system


pressure


system speed of


sound versus temperature should appear


vertical


line


true isothermal


conditions are maintained.


Figures 5-11 and 5-12 illustrate the departure from isothermal

conditions typically encountered in this work.


Figures


5-11


5-12


represent


worst


case


departure


from


isothermal


run


conditions.


general,


drift in temperature during a run averaged approximately 0-1K,


even


this


made


ability


zero


critical


temperature very difficult.


Typically,


a temperature had to


be selected for the constant temperature environment that was


about


tenth


degree


higher


than


temperature


interest


order


assured


being


near


proper


temperature when the system expanded sufficiently to be at the


critical density.


A task that was difficult at best and one


that


required


some


trial


and


error


produce


reasonable















Pressure
for least


isothermal


data


dependence
run.


4.7000




4 6700




4.6400




4.6100


4r.til30


4.5fi0O


1*** F


Sr I i


;1*** I I I


189.400


189.500


189.600


189.700


189.800


Temperature,


189.900
K.


190.000


190.100


I-
Figure 5-11. Pressure dependence on temperature for data run
that deviates most from isothermal conditions.


Speed of Sound Temperature dependence
for least isothermal data run.

220.00 -



170.00 -




-











170.00 -
I


Temperature











vertical lines,


provided the system was truly operating under


isothermal conditions.


A slight drift


in temperature should


produce


a series


shaped


curves


system speed of


sound


decreases,


reaches a minimum and then increases when the


critical


density


approached,


passed and moved away


from.


Figure 5-13


is a plot of system speed of


sound versus system


temperature


:or


data runs.


Inspection shows that every


data run experienced a temperature drift during the course of


experiment.


No amount


time


was


sufficient


to ensure


true


thermal


equilibrium


with


constant


temperature


environment.


Plot


all


data


runs


on


common


set


axes.


-'O O10-''


' } .O0
.00 -









D93QC




*^.~00


C ~ liii. I I 1' I hi IL


1?


"" I


I


I i i I i r r i !1 ii 1 i i i I i i i : I i r i 1 I i 1 i i.i i I 1 I l.i











A three-dimensional


plot


three


system


parameters


clearly


demonstrates


in this experiment.


precision


Figure


5-14


is a plot


technique


employed


of system speed


sound,


system


temperature


system


pressure.


plotted


figure


looks


strikingly


like


head


arrow


pointing


directly


such


striking


critical


point.


indicator


No other technique


location


provides


critical


point.


V niiro


c-li rn n i nn. 1


nl nt- n


c~wst pm


vari a les:


27-0



0m0
"taoo



7 ItoO'0oQ


b id~'a~;


'Phr rs P


5114











Curve Fitting


attain


parameters


critical


point


methane,


is possible to draw the limiting slope asymptotes


on figures 5-9 and 5-13 and identify their intersection as the


critical


point.


However,


such a


technique


is graphical


nature and subject


individual


interpretation.


Since


overall


experimental


technique


based


a mathematical


singularity,


a much more satisfying method of


critical point


identification would involve a fit of the experimental data to


appropriate


mathematical


relationship


from


which


critical parameters may be analytically inferred.

Critical pressure


A model


for describing the behavior of the


latent


heat of vaporization


as a function of temperature was proposed


Stell


Torquoto.40


Their


mathematical


model


subsequent


equation


was


modified by


41,42


Sivaraman


relate the speed of


sound to temperature.


The equation was


derived


from


renormalization


group


theory


Wegner43


later refined by Ley-Koo and Green.


The mathematical model


treats behavior similar to that exhibited by the experimental


data


gathered from


this


work.


The basic equation


relating


speed of sound to temperature is











Where


Speed


Sound,


= Ideal


Constant,


J/ (K)mole


= Critical


Temperature,


= Molecular


Weight,


Kg/mole


= Coefficients


= Reduced


Temperature


IT0)


Equation


critical


utilized


temperature


data


equation


either


contains


side


system


dependent


variables


that


must


found such


that


overall


deviation


function


from


experimental


values


minimized.


The quantity represented by the square root


symbol


in equation


has units


of m/


s and renders


left


side of


equation


unitless.


since


reduced


temperature


units


associated


with


each


variables


right


side


equation


also


unitless.


For work


conducted under


isothermal


conditions,


Eqn.


can be modified to work with pressure instead of temperature.

Wherever the reduced temperature appears, the reduced pressure


can


substituted.


What


constants


should


appear


under


square


root


symbol


left


side


equation


unclear,


they


must


combine


yield


quantity


that


units


s to


assure


that


left


side


new











allow


coefficients


right


side


equation


have


UnJ-zs


of m/


Equation


can


then be


rewritten


= AP


+ A2P$


+ A3P -1 p


Eqn.


A4Pr


+ A2P 2
SA5 P


Equation


only nine system dependent


variables to


be searched instead of


of equation 5-1.


Equation


will not

the nine


yield a value for the critical pressure directly,


syst


em dependent parameters can be found utilizing an


estimated


value


critical


pressure.


critical


pressure


can


then


varied


nine


system


dependent


parameter


s recalculated.


Overall


equation


deviation


can


compared


from


one


iteration


next


until


error


been


reduced


a minimum.


final


value


critical


pressure


taken


that


value


which


produces


minimal


deviation


fitted


equation


from


measured


experimental


results


visible


cross


hatches


figure


5-15


are


actual


experimental


data


a single data


run while


the smooth


line


represents


results


fitting


equation


data.


can


readily


seen


that


fitted


equation


reproduces actual


experimental results very


closely.


The nine


+ APr3







































Figure 5-15. Plot of system speed of sound versus system


pressure


with


fitted


equation


overlaid.


experimental


results,


(chi


squared


from


regression


analysis


was


.9972)


close


examination


exponents


eqn.


indicates


that


However,


another


exponent


already


Thus,


both


terms


can


combined


simplify


equation


two


5-2.


variables


Yet,

are


the

equal


coefficients


magnitude


front


these


opposite


same

sign


suggests


they


cancel


each


other


out.


Consequently,


orni~t- 1 nf


r1 -l II-S


<:i mn1 i f ip


PXnrSS ion


shown


r is


S-


p+s=1


. I I -











Table


5-1.


experimental


System dependent
data.


variables


for equation fitted to


Critical


temperature


begin


search


critical


temperature,


temperature


pressure


can


sought


which


correspond


minimum


plot


speed


system


sound


temperature


each


versus


individual


system


data


pressure


run.


can


constructed


illustrate


dependence


two


system


variables.


Such


a plot


below


critical


point


would be


A~~ ~ ~ a~ 1 r 4 n nt ; F 4- n, nn r r rt~ r c4 a nt r~a ni ryat.Of


i t-rnii ri anH xrnrr


Variable Low Pressure High Pressure
Side Side

A, 2.5834904 m/s 3.5191821 m/s

A, -A4 -A4

A3 300.59068 m/s 286.50994 m/s

A4 -A2 -A2

A, 639132.87 m/s 102031.60 m/s

A6 -77601569 m/s -5604354.6 m/s
a 0.66666667 0.66666667

P -0.25000000 -0.25000000
6 1.2500000 1.2500000

Pc 4.6081493 MPa


:,,, c ,, h F












5-16


plot


system


temperature


versus


system


pressure


where each


individual


point


is at


the minimum system speed of


sound


indi


vidual


data


run.


Plot


corres
for ea


Pressure


ponding
ch data


vs. Temperature


minimum


Speed


Sound


run.


4.7000


4.6750


4.6500


4.6250


Critical point
4.6081 MPa.


189.695 K.


4.6000


189.600


189.775


189.950


190.125


190.200


Temperature,


Figure


press
sound


ure


16.
corres


each


Plot


ponding to


data


system
minimum


temperature
values for


versus
system


system
speed of


run.


line drawn


in figure 5-16


straight


However,


coexistence


curve


between


liquid


and


vapor


states


4-ha~~~ r rAn n ~ i nn -r' ch rn no- t n P~tTIi n ct r~ I'h


all1


F111; cl nnt


C ~ n rn\ h ~J ~ t n 7 M ;











interval.


The temperature value for the critical point lies


somewhere along the line drawn in figure 5-16.

The value for the critical pressure found earlier can be

substituted into the equation of the line shown in figure 5-16


and the critical temperature can be calculated directly.


equation


line


(0.150768MPa/K) T


- 2


3. 9918MPa.


From this equation,


the critical temperature is 189.695K.


Thermodvnamic Surface


The existence of


results


equations that model


in an analytical manner


the experimental


leads to the possibility of


illustrating the thermodynamic surface at the critical point.


Figure 5-17


is an extrapolation of the analytical equations up


through and beyond the critical point of Methane.






















loN
'50


Figure 5-17. Thermodynamic surface of the critical point of
methane.
















CHAPTER 6
DISCUSSION


The Critical Point of Methane


The final critical


values determined from this work are


as follows...


4. 6081MPa.

189.695K


.0015MPa.

.021K


Critical parameters gathered from work published in the

past are listed in table 6-1 along with the present values to


facilitate comparison.


Scrutiny of the values in table 6-1


illustrates


that


this


work


will


rest


controversy

methane.


over

Ample


location


precedence


exists


the

for


critical


higher


point of

critical


pressure


than


latest


value


obtained


Kleinrahm


coworkers.


However,


there is no earlier indication that the


value for the critical


temperature should be as


that


found during the course of this work.


Such


situation


leaves


author


horns


dilemma.


It would be the height of hubris to claim offhand


that this current


piece of work is correct


and all previous


-~~ -r. -: 1L 2 1~ -1 .. -r 2 .. I.-.n ---.


1,,,.,.,,,L




















-. S I j I ~I~~**~A I I I


~I 1 I II I- I I I I'I-I-


-. I- I I-. I I I- I- I~


- b I I -.-- I I_ 1 1 t'"1 I


aI at *I .I SI *I .I -I -I *i II -I I I


I I I I I I I ._I I -











difficult.


possible,


however,


look


potential


systematic errors


search


that might


complete,


exist


in this


precision


this


work.


Once such a


technique


can


favorably


demonstrated


a comparison


this


data


that


obtained


from previous


work


can


be made.


Impurity


reference


cited


just


previous


this


work


table


6-1,


quality


methane


utilized


Kleinrahm


coworkers


was


stated


99.9995


pure.


methane


utilized


for this work was only


99.99% pure.


addition


certain


gases,


most


notably


hydrogen


helium,


could


definitely


affect


results


s experiment


and


yield


inaccurate critical


parameters


Both Hydrogen and Helium gas


typically accompany natural


gas in its native state.


However,


identify


initial


source


possibly


inaccurate


results


inadequate


without


some


type


instrumental


verification.


Such verification was unavailable


time


this


experiments


performance.


System RTD


Keithley


system


196A DMM


was


used


purchased


measure


precalibrated


resistance


from











been


obtained less than six months


previous


to performance of


this experiment,


calibration was deemed unnecessary.


Also,


precision


resistor


was


required,


unavailable.


precaution,


Keithley


196A


was


checked


against


precalibrated


RTD.


Resistance


values


corresponding


temperature


values


were


supplied


manufacturer.


DMM


was


properly


calibrated,


should always


produce


resistance


values


that


matched


resistance


values


supplied by


manufacturer


specific


temperature.


precalibrated


value


was


triple


checked


point


against


of water.


known


temperature


difference


between


resistance


reading


and


stated


manufacturer'


resistance


value


was


treated


offset


throughout


experiment.


reconcile


necessity


the DMM to actual


using


temperature


resistance


values might


offset


indicate


that


DMM


further


systematic


errors


that


went


undetected.


Finally,


physical


location


system


was


above


spherical


resonator.


flow


of methane


was


into


the bottom of the resonator and out


the top.


Placement of the


system


close


point


which


methane


was


being examined by acoustic excitation was essential.


However,


constant


temperature


environment


was


kept


cool











environment


directly


through


blades


circulating


fans.


Potentially,


temperature


directly


adjacent


circulating


fans


might


have been


colder


than


rest


environment.


fans


were


located


near


constant


temperature


environment


one


side


and


near


bottom on the other side to provide maximum


cross


circulation.


possible


that


location


system


near


top of the


constant temperature environment might have been an


error.


shield


physical


barrier


was


placed


between


circulating


fans


and the RTD


shroud during all


experiments


help alleviate


this potential


error


, but


it must be mentioned


order


Heise


to be


Pressure


thorough.


Gauge


The

absolute


Heise

dead w


pressure


eight


gauge


pressure.


was ca

However,


librated


against


calibration


did not


extend


into


region


pressure


necessary


this


experiment.


It was assumed that extrapolation of the pressure


calibration to higher pressures was


legitimate because of the


extreme linearity


of the calibration plot


lower pressures.


Such


assumption


have


been


appropriate.


Still,


great


care


was


exercised


calibration


Heise


pressure


gauge


well


the measurement


devices.