a dynamic response model for pressure sensors in continuum and
TRANSCRIPT
NASA Technical Memorandum 4728
A Dynamic Response Model for Pressure Sensors in Continuum and High Knudsen Number Flows with Large Temperature Gradients
January 1996
Stephen A. Whitmore, Brian J. Petersen,
and David D. Scott
NASA Technical Memorandum 4728
National Aeronautics and Space Administration
Office of Management
Scientific and Technical Information Program
1996
Stephen A. Whitmore
Dryden Flight Research CenterEdwards, California
Brian J. Petersen
UCLA School of Engineering and Applied SciencesLos Angeles, California
David D. Scott
Lawrence Livermore National LaboratoriesLivermore, California
A Dynamic Response Model for Pressure Sensors in Continuum and High Knudsen Number Flows with Large Temperature Gradients
A DYNAMIC RESPONSE MODEL FOR PRESSURE SENSORS INCONTINUUM AND HIGH KNUDSEN NUMBER FLOWS WITH
LARGE TEMPERATURE GRADIENTS
Stephen A. Whitmore*
NASA Dryden Flight Research CenterEdwards, California 93523
Brian J. Petersen†
UCLA School of Engineering and Applied ScienceLos Angeles, California 90024
David D. Scott††
Lawrence Livermore National LaboratoriesLivermore, California 91283
Abstract
This paper develops a dynamic model for pressuresensors in continuum and rarefied flows withlongitudinal temperature gradients. The model wasdeveloped from the unsteady Navier-Stokes momentum,energy, and continuity equations and was linearizedusing small perturbations. The energy equation wasdecoupled from momentum and continuity assuming apolytropic flow process. Rarefied flow conditions wereaccounted for using a slip flow boundary condition atthe tubing wall. The equations were radially averagedand solved assuming gas properties remain constantalong a small tubing element. This fundamental solutionwas used as a building block for arbitrary geometrieswhere fluid properties may also vary longitudinally inthe tube. The problem was solved recursively starting atthe transducer and working upstream in the tube.Dynamic frequency response tests were performed forcontinuum flow conditions in the presence oftemperature gradients. These tests validated therecursive formulation of the model. Model steady-state
* Vehicle Dynamics Group Leader, Aerodynamics Branch,Member, AIAA.
† Graduate Student, Student Member, AIAA.††Engineer.Copyright 1996 by the American Institute of Aeronautics and
Astronautics, Inc. No copyright is asserted in the United States underTitle 17, U.S. Code. The U.S. Government has a royalty-free license toexercise all rights under the copyright claimed herein for Governmentalpurposes. All other rights are reserved by the copyright owner.
behavior was analyzed using the final value theorem.Tests were performed for rarefied flow conditions andcompared to the model steady-state response to evaluatethe regime of applicability. Model comparisons wereexcellent for Knudsen numbers up to 0.6. Beyond thispoint, molecular affects caused model analyses tobecome inaccurate.
Nomenclature
A temperature profile label, ~ ambient
tube cross-section area, in.
ac alternating current
A/D analog-to-digital conversion
constant of integration, frequency variable
B temperature profile label, ~ 350 °F
constant of integration, frequency variable
C temperature profile label, ~ 500 °F
c sonic velocity, ft/sec
specific heat at constant pressure, ft-lbf/[lbm °R]
D temperature profile label, ~ 650 °F
d tube diameter, in.
D/A digital-to-analog conversion
dB decibel
dc direct current
Tmax
Ac
Aω
Tmax
Bω
Tmax
Cp
Tmax
minimum frequency in phase-modulated wave, Hz
i general index
j
zeroth order Bessel function
first order Bessel function
second order Bessel function
K polytropic density proportionality constant
polytropic temperature proportionality constant
L tube length, in.
lsb least significant bit
M(x) constant of integration, energy equation
n number of computational nodes
N(x) constant of integration, momentum equation
number of harmonics in phase-modulated wave
P pressure, psf
pressure at cold end of tube, psf
pressure at hot end of tube, psf
pressure at transducer, psf
Prantl number
pressure at surface, psf
mean pressure in tubing, psf
R tube radius, in.
r radial coordinate, in.
universal gas constant, ft-lbf/[lbm °R]
broadband waveform
SPL sound pressure level, dB
T temperature, °R
t time, sec
temperature at cold end of tube, °R
temperature at hot end of tube, °R
temperature at transducer, °R
wall temperature, °R
temperature at surface, °R
thermocouple readings along tube, °F
U longitudinal velocity, ft/sec
radial average of longitudinal velocity, ft/sec
creep flow velocity at tubing wall, ft/sec
slip velocity at the wall
steady-state flow velocity, ft/sec
V entrapped transducer volume, in3
effective volume of a model node
x longitudinal coordinate, in.
z independent variable for Bessel equation
shear wave number
ratio of specific heats
propagation factor
frequency resolution, Hz
ratio of slip distance to mean free path
thermal conductivity, lbf/[sec/°R]
slip distance, in.
Knudsen number
rarefied flow correction term in momentum equation
Knudsen number based on mean flow properties
mean free path of the fluid molecules, microns
dynamic viscosity, lbm/[ft/sec]
local steady-state bulk viscosity in the tubing, lbm/[ft/sec]
polytropic expansion parameter
the constant
density, lbm/ft3
mean density, lbm/ft3
rarefied flow static response parameter
radian frequency, 1/sec
velocity gradient at the wall
f0
1–
J0
J1
J2
K'
Nh
Pcold
Phot
PL
Pr
P0
P0
Rg
Sigt
Tcold
Thot
TL
Tw
T0
T1...6
Uavg
Ucreep
Uslip
Uss
Ve
α
γ
Γp
δf
ε
η
ϑ
κ
K 1.7725 Rg µR--- T 495.7+( )
p---------------------------- ≈
κp
κ0
λ
µ
µ0
ξ
π π 3.141726≈( )
ρ
ρ0
Ψ
ω∂U∂rw---------
2
Introduction
With development of advanced hypersonic vehicleconcepts, reliable measurement of onboard trajectoryparameters from pneumatic sensors is highly desirable,but measurement of aerodynamic properties onhypersonic vehicles presents formidable challenges. Thehostility of the sensing environment precludes intrusioninto the flow, and measurements must be obtained viaremote sensors. This hostile environment requires usingsizable lengths of pneumatic tubing to transmit pressurefrom the surface to the remotely located transducer. Forhypersonic conditions, nominal spectral distortion andacoustical resonance affect the measurements. Inaddition, large temperature (T) gradients induced byboundary-layer heating induce molecular effects wheregas molecules adjacent to the tube wall creep from thecold end of the tube to the hot end. Furthermore, underlow pressure conditions, such as those experienced atvery high altitudes, the tube flow becomes so rarefiedthat the fluid slips at the tube wall.
To date, no theoretical model describing influenceof these rarefied flow phenomena on pressuremeasurements is available in the scientific literature. Togenerally quantify the dynamic behavior of pressuresensors for these hypersonic conditions, NASA DrydenFlight Research Center, Edwards, California, initiatedresearch to develop an accurate frequency responsemodel that is mathematically invertible. That is, giventhe measured pressure at the pressure sensor, the modelcould predict the pressure input which occurred at thesurface. This invertible model could compensate forpneumatic measurement distortions which cannot bemitigated by the layout of the pneumatic hardware.
This paper describes a general dynamic responsemodel for pressure sensors and applies to continuumand rarefied flow conditions. The model allows for largetemperature gradients on the order of 1000 °R/ft with amaximum Knudsen number, of approximately 0.60.The model was verified using steady-state and dynamiclaboratory experiments. Test results and regimes ofapplicability are also presented. Finite differencemethods were not desirable for this application. Instead,the equations of energy and motion are decoupled andreduced to a one-dimensional boundary value problem.The boundary value equations are solved assuming thatalong a small element gas properties remain constant,and a fundamental solution is developed for this small
element. Then, the fundamental solution is used as abuilding block for a recursive solution method whichallows for complex geometries where fluid propertiesand tubing geometry can vary longitudinally. Theproblem is solved recursively starting at the transducerend and working toward the surface end of the tube.Using these recursive formulae, solutions for arbitrarygeometries and longitudinal temperature profiles can beconstructed. The resulting model is fully invertible.
Background
For full continuum flow which occurs at moderatepressure levels, fluid viscosity causes the gas to stick atthe wall, resulting in the classical no-slip boundarycondition. On the other hand for rarefied flowconditions, molecular effects become important, and theno-slip boundary condition is no longer valid (ref. 1).For rarefied flow conditions, fluid elements do not stickto the wall as they would in continuum flow. Instead,fluid elements slip along the wall, resulting in a flowregime that is referred to as slip flow. The magnitudes ofthe molecular effects are proportional to ratio of themolecular mean free path to the characteristic scalelength of the flow — Knudsen number. Values ofKnudsen numbers less than 0.01 indicate that the flowconditions are continuum and that molecular effectsmay be ignored. Values between 0.01 and 1.0 indicatethe slip flow regime where the flow has elements ofcontinuum and molecular dynamics. Values exceeding1.0 indicate a free-molecule flow regime, andcontinuum affects can be ignored.
The mean free path of the fluid molecules, , is theaverage distance that each fluid particle travels betweensuccessive collisions with other fluid particles. Ifcharacteristic scale of the system were the tube radius,then the Knudsen number can be approximated by theexpression given in equation (1) (ref. 1).
(1)
where is the constant, is the dynamic viscosity, is the universal gas constant, R is the tube radius,
T is the temperature, and P is the pressure.
Clearly, Knudsen number is an inverse function ofpressure. Thus, rarefied flow phenomena are inherentlyassociated with high Knudsen numbers. Forconventional aeronautical applications, the Knudsen
κ,
λ
κ Rgπ µR---
T P
---------≈
π µRg
3
numbers are always below 0.01. For hypersonic andorbital applications, Knudsen numbers from 0.05 to0.20 can be obtained along the flight profile at very highaltitudes. Figure 1 illustrates these flow regimes.
The problem of predicting tube flow dynamics hasbeen studied extensively. For nonrarefied, constanttemperature conditions, Iberall developed a spectraltechnique for predicting response lags for sinusoidalinputs and lightly damped configurations (ref. 2). Lambadapted the work of Iberall to predict step-input risetimes for highly damped configurations (ref. 3). Lamb’stheory was also applied to predicting steady-state delaysfor constant ramp inputs. Schuder, et al., (ref. 4) andHougen, et al., (ref. 5) developed closed form frequencydomain solutions for simple tubing geometries andconstant wall temperatures. Bergh, et al., (ref. 6)extended the analyses of references 4 and 5, to develop arecursion formula for complex geometries. The workdetailed in reference 6 is the state-of-the-art forpredicting tubing responses to constant walltemperatures, and continuum flow. Tijdemanextends the model boundary condition to allowfor high-speed surface cross flow (ref. 7). He alsopresents a succinct summary of existing tube responsetheories (ref. 8).
Parrot, et al., investigated the dynamic transmission ofsound in a simple geometry tube which was subjected tovery large temperature gradients (ref. 9). These testswere performed for ambient pressure levels, and rarefiedflow effects were not considered. Knudsen (ref. 10) andKennard (ref. 1) investigated tube flow for rarefiedconditions with large temperatures. These analyses,however, have been performed only for simplegeometries and for steady-flow conditions. A review ofthe literature did not reveal an unsteady response model.References 1 and 10 present excellent overviews oftheoretical and empirical results for steady-state tubeflows in rarefied conditions.
Along an unequally heated gas boundary, Maxwell’skinetic theory (ref. 1) predicts that gas moleculesoriginating in the hot region of the tube have higherkinetic energy than molecules originating from the coldregion. As a result, such molecules recoil more stronglythan molecules from the cold side of the tube. The netresult is that the gas acquires a longitudinal momentumin the hotter direction. This net momentum gain causesthe gas molecules at the wall to creep from the cold endto the hot end of the tube. To balance this creep, gas
molecules in the center of the tube must migrate towardthe colder end of the tube. This opposing flowequilibrium results in establishment of a steady-statepressure gradient. The cold region of the tube has alower pressure than the hot region, and no net cross-sectional flow exists in the tube.
In his analyses, Maxwell determined that in the freemolecular limit the normalized ratio of the creep-induced pressure gradient was one-half of thenormalized temperature gradient. For example,
(2)
where is the induced longitudinal pressuregradient, P, is the nominal pressure in the tube, is the longitudinal temperature gradient, and T is thenominal temperature in the tube. For conditions whichlie somewhere between the free molecular regime andcontinuum flow, the pressure gradient induced bylongitudinal temperature gradients is less than one-halfand is a strong function of Knudsen number (refs. 1and 10).
For slip flow conditions, the primary molecular effectis the fluid movement at the wall boundary. The fluidvelocity at the wall boundary can be decomposed intotwo parts: slip velocity and thermomolecular creepvelocity. The slip velocity, (ref. 1) is proportionalto the shear stress at the wall, and for laminar flow maybe written in terms of the velocity gradient as
(3)
where U is longitudinal velocity, and r is the radialcoordinate.
The parameter, , is referred to as the slip distanceand is dimensioned in units of length. The slip distanceis on the order of the mean free path of the flow (ref. 1).The ratios of slip distance to mean free path for variouschannel materials and gases are tabulated in references 1and 9. For the flow of air over machined brass or steel,the ratio is 0.995; for air flowing over glass, the ratiois 1.24.
The molecular creep velocity is one of the morepeculiar phenomenon which occurs at low pressure.
Tw,
∂P∂xP------------
12---=
∂T∂xT------------
∂P ∂x⁄∂T ∂x⁄
Uslip,
Uslip ϑ ∂U∂rw---------–=
ϑ
4
Kennard shows that the creep velocity is directlyproportional to the longitudinal temperature gradientand inversely proportional to the local pressure (ref. 1).Its magnitude can be approximated by the expression
(4)
where and are longitudinal average viscosityand pressure in the tubing. At the wall boundary, thevelocity is the sum of the two terms. For example,
(5)
For slip flow conditions, the condition givenby equation (5) replaces the traditional no-slip,
, boundary condition used in continuumfluid mechanics. Other than this modification, theclassical equations of fluid motion apply in this flowregime (refs. 1 and 11).
Mathematical Analysis
This section presents the mathematical analyses usedto develop the rarefied flow dynamic model. Theboundary value equations describing the pressure wavepropagation in the tube are presented first. Next, arecursive solution method for these boundary valueequations is stated. Finally, the steady-state behavior ofthe frequency response model is analyzed. AppendixesA, B, and C present detailed development of allmathematical analyses.
Derivation of the Boundary Value Equations
The model is derived from the unsteady, three-dimensional Navier-Stokes equations which arelinearized using a small perturbation assumption. Theenergy equation is decoupled from the equations ofmomentum and continuity, assuming the longitudinalwave expansion process within the tube is polytropic(refs. 1–7, 11, and 12). For a polytropic process, therelationship between pressure, temperature, and densityis described by the simple model
(6)
where
Limiting values for are given by where corresponds to an irreversible isothermal
expansion process, and corresponds to acompletely reversible isentropic expansion process.Using the polytropic flow assumption allows decouplingthe energy equation from the equations of momentumand continuity without loss of generality. Appendix Apresents the variation of as a function of inputfrequency and the fundamental flow parameters. Themomentum equation is integrated to give the local flowvelocity in terms of the longitudinal pressure gradient.The slip flow boundary condition (eq. (5)) is used tosolve for the constant of integration.
The result is averaged over the cross-section of thetube to give a radially averaged flow equation. Theresulting equation is coupled with the radially averagedcontinuity equation to develop a wave equation whichdescribes the pressure propagation in the tube forrarefied flow conditions. Similar arguments are used todevelop a downstream longitudinal boundary condition.The upstream pressure is assumed to be a prescribedinput. The resulting boundary value equations,
wave equation
(7)
downstream boundary condition
(8)
Ucreep34---
µ∂T ∂x⁄ρ0T x( )-------------------- 3
4---
µ0Rg
P0------------
∂T∂xw---------≈=
µ0 P0
U x R t, ,( ) Uslip Ucreep
ϑ ∂U∂r-------–
34---
µ0Rg
P0------------ ∂T ∂x⁄+≈
+=
U x R t, ,( ) 0=
P K ρξK' T
ξ ξ 1–( )⁄= =
= pressure
= proportionality constant for density
= proportionality constant for temperature
= polytropic expansion parameter
= density
= temperature
P
K
K'
ξ
ρ
T
ξ 1 ξ γ< <ξ 1=
ξ γ=
ξ
∂2P x( )
∂x2
----------------- ω Γ p
c------
2
P x( )=
∂P∂x
x L=
------------------- ω2Γ p
2
c2
------------- VAc------ PL–=
P 0 ω,( ) P0 ω( )≡
5
propagation velocity
(9)
and upstream pressure input, , are prescribed.(See appendix A.)
(10)
where
The propagation factor, given by equation (10) isa new result not presently available in the scientificliterature. This factor accounts for molecular effects andfrictional damping in the tube and is a generalization ofthe work presented in references 6 and 7. Parameters and are the Bessel functions (ref. 13) of the zerothand first orders.
Boundary Value Equations Solution
The boundary value equations are solved in thefrequency domain. Temperature and gas properties areassumed to remain constant along the length of the tubeto give a fundamental solution where the complexspectra are given as a function of the sensor geometry,the frequency of the input sinusoid, and the propagationfactor. This solution is detailed in appendix B.
(11)
This fundamental solution for constant temperatureand tube radius is used as a building block for complexgeometries in which the wall temperature, fluidproperties, and tube geometry vary longitudinally. Forlongitudinal variations within the tube, the problem issolved recursively starting at the transducer end andworking toward the surface (external) end of the tube.As developed in appendix B, the solution at the ithnode is
(12)
where the effective volume, accounts for theentrapped volume at the ith node plus the impedance ofthe downstream tubes and volumes. The general end-to-end frequency response is given as the complex productof the frequency responses at the individual nodes.
(13)
= transducer volume
= sonic velocity
= tube cross-sectional area
= tube length
= pressure at the transducer
= surface pressure
= rarefied flow correction factor for the bulk viscosity
= longitudinally averaged velocity
= radian frequency
= shear wave number
= longitudinal pressure gradient
= mean pressure in the tubing
Uavg x( )
∂P∂x------
jω ξγ-- Γ p
2 ρ0
--------------------------------- =
P0 ω( )
Γ p
γξ--
J0 α[ ] ϑ αR--- J1 α[ ]–
κ p J2 α[ ] ϑ αR--- J1 α[ ]+
jω 34---
µ0
P0------
ξ 1–ξ
--------- 2α--- J1 α[ ]+
----------------------------------------------------------------------------------------------------------------------------≡
V
c
Ac
L
PL
ρ0
κ p
Uavg
ω
α
∂P∂x------
P0
Γ p,
J0J1
PL ω( )
P0 ω( ) 1
ωΓp Lc---
VωΓp
Acc--------------- ωΓp
Lc---sinh+cosh
---------------------------------------------------------------------------------------------=
Pi ω( )
Pi 1– ω( )
ωΓpi
Li
ci----- ω
Vei
Aci
------- Γ pi
ci------- ωΓpi
Li
ci-----sinh+cosh
-----------------------------------------------------------------------------------------------------------=
Ve,
PL ω( )P0 ω( )----------------
P1 ω( )P0 ω( )---------------=
P2 ω( )P1 ω( )--------------- …
Pn 1– ω( )Pn 2– ω( )----------------------
Pn ω( )Pn 1– ω( )----------------------
1
ωΓpi
Li
ci----- ω
Vei
Aci
------- Γ pi
ci-------- ωΓpi
Li
ci-----sinh+cosh
------------------------------------------------------------------------------------------------------------
i 1=
n
∏=
6
Equations (11) through (13) allow for a finite-elementsolution of the boundary value problem. Theseequations are used to generate the frequency responsesolutions for complex geometries or for longitudinaltemperature distributions. If molecular effects areignored, and a constant temperature profile is assumed,these equations are mathematically identical to therecursion formulae developed in references 6 and 7.
Steady-State Response of the Dynamic Model
Maxwell’s analysis predicts that in the presence oflarge temperature gradients and rarefied flow, theequilibrium pressure gradient in the tube is nonzero.The model (eqs. (10) – (13)) exhibits a similar steady-state behavior. Equilibrium behavior of the model forrarefied flow conditions is best understood by looking atthe momentum equation at a given longitudinal cross-section. The normalized steady pressure gradient can bewritten as a function of Knudsen number andnormalized temperature gradient.
(14)
(See appendix C.) In equation (14), is the ratio of theslip distance to the mean free path and, for this analysis,can be assumed to be unity. Equation (14) is extremelyimportant because Maxwell predicts that in the free-molecule limit (Knudsen numbers approaching infinity)
(15)
Clearly, equation (6) does not approach a limit.Therefore, the model has an upper boundary for whichthe slip flow assumptions are valid. Because it isextremely difficult to conduct controlled dynamicexperiments under rarefied flow conditions, steady-statebehavior of the model is the only feasible means ofevaluating the validity and range of applicability for theslip flow assumptions used in deriving the dynamicmodel. Empirical validation of the model is describednext.
Experimental Apparatus and Procedures
The assumptions used in deriving the model and theflow regimes to which the model applies were evaluatedusing a series of laboratory tests. First, dynamic
frequency response tests were performed for continuumflow conditions at room temperature and in the presenceof large temperature gradients. Results of these testsdemonstrated the validity of the polytropic energyanalysis and the recursive formulation for temperaturegradients in the tubing. Next, steady-state response testswere performed for rarefied flow conditions. Thesetests were used to verify the slip flow assumptionsused in deriving the boundary value equations andto establish a regime of validity for the model. Resultsfrom these experimental tests are compared to analyticalpredictions in the Results and Discussion section.
Dynamic Frequency Response Tests
Frequency response measurements were gatheredusing a test plate mounted at the end of the soundchamber. Figure 2(a) presents a schematic of the testconfiguration, and figure 2(b) shows an overview of thetest equipment layout. Reference sound pressure levelsimpinging on the plate were measured by a constantcurrent piezoelectric microphone mounted flush to theplate. The response of a test configuration, whichconsisted of a flush surface port and a section of brasstubing, was measured by an identical microphonemounted in a housing at the end of the tubing.
Frequency response was evaluated by comparing theoutput of the test microphone to the output of thereference microphone. For these tests, a broad-bandwave form was generated by a microcomputer outfittedwith a 12-bit digital-to-analog (D/A) conversion board,amplified with commercial stereo equipment, and usedto excite a large speaker inserted in an anechoicchamber. The speaker is shown in the sound chamber infigure 2(c). Speaker volume was controlled using avoltage attenuator on the output voltage from themicrocomputer. By changing the speaker sizes andoutput roll off, frequency ranges from approximately0.50 to 2000 Hz could be accurately evaluated.
High-temperature gradients were induced by a heatermade from a 3/4-in. diameter aluminum rod heated withelectrical resistance heating tape. The rod was boredwith a hole its entire length, and the section of brasstubing to be evaluated was press-fit into the hole. Thetemperature of the heating rod was regulated using atemperature controller and a feedback thermocouple.One end of the brass tube was soldered flush to thesurface port in the test plate, and the other end was fittedto the microphone housing. Copper-constant (Type T)
∂P∂x------
P-------
6κ02
π 1 4 ε κ0+
---------------------------------
∂T∂x------
T-------≈
ε
∂P∂x------
P-------
∂T∂x------
T-------÷ 1
2---=
7
thermocouples (TC) were used to sense the temperatureat the surface port, the microphone housing, and fourpoints along the length of the tube. Thermocouples werejoined to a single electronic reference junction, and theirreadings were selectable using a rotary switch. In theranges tested, the estimated accuracy of thethermocouple measurements was approximately ± 2 °F.These ranges were based on the manufacturer’sspecifications. Figure 2(c) also shows the heaterconfiguration and the test plate arrangement.
Output signals from the reference and testmicrophones were amplified and sampled at 24 kHz bya 16-bit analog-to-digital (A/D) conversion board in themicrocomputer. Direct current (dc) offsets in themicrophone outputs were removed by alternatingcurrent (ac) coupling the microphone outputs to give aminimum response frequency of approximately0.05 Hz. The flat frequency responses of themicrophones and signal conditioning extended to wellbeyond 10 kHz. At nominal sound volume levels,measurements showed that the sound pressure level(SPL) in the chamber was approximately 145 dB(7.5 psf). This SPL is well within the linear responseregions of the reference and test microphones. In thelinear range these microphones have an unamplifiedresponse sensitivity of 10.4 volt/psf. Using amplifiergains settings of 10 and a full-scale A/D range of± 1 volt, the nominal resolution of the microphone leastsignificant bit (lsb) was approximately 0.0029 psf/lsb.
The broad-band wave form used to excite the speakerwas generated by a nonlinear phase-modulated cosineseries of the form
(16)
where is the minimum frequency in the wave form, is the spacing between harmonics, is the number
of harmonics in the wave form, is time, and isthe maximum excited frequency. The nonlinear phasemodulation ensures that energy is distributed uniformlyin the time and frequency domains (ref. 14) and that thewave form will be physically realizable.
Figures 3(a) and 3(b) show the time history andspectra of a sample wave form. Large sample runs(typically 100,000 data points) were taken for eachtest, and an ensemble of coarse transfer functions
was evaluated using a fast Fourier algorithm with a4096-point data window. The coarse transfer functionsresulting from each data window along the time historywere ensemble averaged for the entire data record.Ensemble averaging helps to mitigate the affects ofresolution and random measurement errors andproduces a clean transfer function output.
For each data run, ambient pressure levels wererecorded with a hand-held manometer, and a baselinedata set at ambient temperature levels was taken. Theheater was turned on with the required setting selectedon the temperature controller, and the system wasallowed to stabilize. The frequency response data wereobtained, and the temperature readings at each of the sixthermocouples was recorded. The temperature was thenraised to the next condition and allowed to stabilize.After data at the maximum temperature which could beobtained by the system, approximately 650 °F, wasrecorded, the system was allowed to cool. Next, the testsat lower temperature settings were repeated.
Steady-State Response Tests for Rarefied Flow Conditions
Figure 4(a) shows the apparatus layout for the steady-state response tests. These tests were used to verify thesteady-state response of the analytical model forrarefied conditions and to evaluate the upper limit ofKnudsen numbers for which the model is valid. Asmentioned in the Background section and appendix C,the model theory predicts that in the presence of largetemperature gradients and rarefied flow conditions, gasadjacent to the tube wall creeps from the colder regionto the hotter region. The result is an opposing flowwhich establishes a steady-state pressure gradientwithin the tube, with the cold region of the tube readinglower temperatures than the hot region. These testsreproduced those conditions.
The steady-state response tests were performed in anevacuated vacuum oven. Here, 3/4-in. diameteraluminum rods were bored with holes, and anassortment of brass tubes of varying diameters andlengths was press-fit into the holes. As the oven washeated, the aluminum rods provided a thermal mass todistribute the heat evenly along one end of the tubing.Type T thermocouples bonded to each end of the tubewere used to sense the absolute temperatures and thetemperature gradient along the tube. As before, thethermocouples were joined to a single electronic
µ
Sigt 2π f0 iδf+( )ti3πNh-------+cos
i 0=
Nh
∑=
f0δf Nh
t Nhδf
8
reference junction, and their readings were selectableusing a rotary switch. Accuracies were similar to thosevalues obtained in the frequency response tests.Thermocouple wire was passed from the vacuumchamber to the thermocouple reference panel using ahermetically sealed thermocouple fitting on the back ofthe oven.
The heated end of the tube was open to the ovenchamber, and the cold end was hermetically bonded to acompression fitting that allowed access to the tube fromoutside of the evacuated vacuum chamber. The chamberpressure was measured using a highly accurate Verniermanometer, and the pressure differential in the tubingwas measured using a differential McLeod gauge(ref. 15). The oven vent was branched off to themanometer and to the reference side of the McLeodgauge and pressure valve was used to isolate the twoinstruments when readings were being taken. A close upof the test configuration showing the attachedthermocouples and the pressure fittings is shown fromthe front view in figure 4(b) and from the rear view infigure 4(c).
Tests were conducted by first recording the zerodifferential pressure in the tube at ambient temperatureand pressure. The heater was turned on with therequired setting selected on the temperature controller.Next, the system temperature was allowed to stabilize.Then, the chamber was evacuated to the approximatedesired pressure level, and the system was sealed. Thehot and cold end temperatures were recorded using thethermocouples. At this point, the chamber pressure wasrecorded using the Vernier manometer, and thedifferential pressure in the tubing was recorded usingthe McLeod gauge. At the end of each data point, thetemperature setting was maintained constant. Inaddition, chamber pressure was adjusted to the newdesired value. The system was allowed to stabilize, anda new set of readings was taken.
For each temperature setting, approximately30 pressure test points were recorded, starting at thelowest pressure and working toward higher pressures.At the end of each set of runs, the system was ventedand allowed to cool to ambient temperatures. Then, anew zero differential pressure reading was taken. Thepre- and post-test zero readings were used to correct thedifferential pressure measurement for bias offsets in theMcLeod gauge. Standard accuracy for a Vernier
manometer is on the order of 10-25 microns of mercury(0.030-0.080 psf). The accuracy of the differentialMcLeod gauge is on the order of 5-10 microns ofmercury (0.015-0.030 psf) (ref. 15).
Results and Discussion
Results of the frequency response tests are describedfirst. Comparisons to the analytical model for selectedgeometries and temperature profiles are presented. Next,the results of the steady-state response tests arepresented. These data were parameterized as a functionof Knudsen number and compared to the predictions ofthe analytical model. From the comparisons, a range ofvalid Knudsen numbers for the model has beenestablished.
Frequency Response Tests
Sixteen data runs were performed. There were fourtemperature profiles for each of the two tubegeometries. Each test was repeated twice. All of the testswere performed using a broad-band wave form withspectral energy from 10 to 4000 Hz. Table 1 presents thefrequency response test matrix, including the tubegeometry and the temperature readings for each of thesix thermocouples bonded to the tube. For these tests,the maximum attainable heater temperature wasapproximately 650 °F. This configuration resulted in amaximum temperature gradient of approximately1300 °F/ft.
Temperature readings from the repeated data runswere averaged and interpolated to give idealizedtemperature profiles along the length of the tube. Theseprofiles, labeled A, B, C, and D, are presented infigures 5(a) and 5(b) for the 0.066- and 0.033-in.diameter tubes. Using these idealized temperatureprofiles, the theoretical frequency responses of the testgeometries were evaluated using the recursive formulaof equation (13).
The computations were basically insensitive to thenumber of grid points. This insensitivity is illustrated infigure 6 where the frequency response of the 0.066-in.diameter tube was evaluated, assuming temperatureprofile D. These computations were performed with 5,10, 20, 50, and 100 equally spaced nodes. Beyond10 elements, little difference exists in the computations.Beyond 20 elements, the solutions are virtuallyidentical. Thus for this analysis, 20 solution elements
9
were used. The resulting calculations are shown infigure 7(a) for the 0.066-in. diameter tube andfigure 7(b) for the 0.033-in. diameter tube fortemperature profiles A, B, and C.
The overall effect of increasing temperature gradientsis an increase of the phase delay of the response and ashift of the spectral harmonics to higher frequency andlower magnitude. This effect is verified extremely wellby the data. The results are presented in figures 8 and 9for the 0.066- and 0.033-in. diameter geometries. Here,the model calculations are overplotted against thetransfer function data averaged from the repeated runsfor the various temperature profiles of figure 5. Theagreement is excellent for all of the cases. Up to theapproximately 2000-Hz limits of the data, the frequencyresponse is predicted to within the noise limits of thebasic measurement. That is, the locations of theharmonics are predicted to within 1-2 Hz, and thespectral magnitudes are predicted to within 2 dB alongthe entire frequency band. Clearly, the energy analysisand recursive formulation are entirely valid for thetemperature ranges presented.
Steady-State Response Tests for Rarefied Flow Conditions
These steady-state response tests were performed toassess the upper limit of Knudsen number for which therarefied flow terms in the model are valid. For thesteady-state tests, tube diameters from 0.092- to0.014-in., temperature gradients as high as 950 °F/ft,and chamber pressures as low as 100 microns ofmercury (0.28 lbf/ft2) were tested. The resultingKnudsen numbers varied from zero to approximately10. Table 2 presents the test matrix which wasinvestigated.
Rearranging equation (14) to collect Knudsen numbergives
(17)
Equation (17) suggests a manner to display the resultsof the steady-state response tests. Approximating thederivatives in equation (17) by differences yields
(18)
These data can be collapsed to a single curve byplotting the rarefied flow static pressure parameter, ,against Knudsen number averaged over the hot and coldends of the tube. These results are plotted in figure 10along with equation (17) evaluated using 1.0 for theproportionality constant, .
For pressures below 350 microns (0.28 lbf/ft2), thevacuum oven chamber pressure was difficult tomaintain, and the data are somewhat suspect. However,these data appear to approach the free-molecule limit of0.5 (eq. (2)). The comparison to the model is excellentfor Knudsen numbers up to approximately 0.6. Becauseequation (17) was derived directly from the fundamentalsolution of the dynamic model (eq. (11)), the modelappears valid for most of the slip flow regime. For
, free-molecule affects dominate, and the modelrapidly diverges from the data. This Knudsen number isthe upper boundary on the model’s usefulness. Foraeronautical applications, this Knudsen number occursonly under near-orbital conditions (fig. 1).
Concluding Remarks
Measurement of aerodynamic properties onhypersonic vehicles presents formidable challenges. Thehostility of the sensing environment disallows intrusioninto the flow. For this reason, measurements must beobtained through remote sensors. In addition, sizablelengths of pneumatic tubing must be used to transmitpressure from the surface to the remotely locatedtransducer. Because pneumatic measurements arenecessary to compute vital flight mechanics parameters,such as angle of attack, dynamic pressure, and Machnumber, or to evaluate surface pressure distributions, itis essential that the dynamic behavior of tubingtransducer measurement configurations be wellunderstood for hypersonic flight conditions. Theseconditions include high surface temperature gradientsand rarefied flow.
This paper develops a general dynamic responsemodel for pressure sensors in high Knudsen numberflow with large temperature gradients. The modelapplies to continuum and rarefied flow conditions andallows large temperature gradients within the pneumatictubing. The sensor response model is developed fromthe Navier-Stokes equations and linearized by smallperturbations. It decouples the energy equation by
TP---
∂P ∂x⁄∂T ∂x⁄-----------------
6κ02
π 1 4 ε κ0+( )---------------------------------≈
∂P ∂x⁄P
-----------------
∂T ∂x⁄T
---------------------------------
Phot Pcold–
12--- Phot Pcold+( )-----------------------------------------
Thot Tcold–
12--- Thot Tcold+( )----------------------------------------
----------------------------------------- Ψ≡≈
Ψ
ε
κ 0.6>
10
assuming that the wave expansion in the tube ispolytropic.
The model is converted to a one-dimensionalboundary value problem by radially averaging flowproperties. The boundary value equations are solved inthe frequency domain, assuming that the gas propertiesremain constant along the length of the tube. Thisfundamental solution is used as a building block forcomplex geometries in which the fluid properties in thetube vary longitudinally. The problem is solvedrecursively starting at the transducer end and workingtoward the surface end of the tube. Using the recursiveformula, solutions for arbitrary geometries andlongitudinal temperature profiles can be constructed.
The steady-state behavior of the model is analyzedby applying the final value theorem to therecursive equation. The resulting expression isnondimensionalized and written as a function ofKnudsen number. The steady-state response function isused to evaluate the regime of applicability of thedynamic model.
The assumptions used in deriving the model and theflow regimes to which the model applies were evaluatedusing a series of dynamic and steady-state laboratorytests. Dynamic frequency response tests were performedfor continuum flow conditions and temperaturegradients as large as 1300 °F/ft. Steady-state responsetests were performed for rarefied flow conditions withchamber pressures as low as 100 microns of mercury(0.28 lbf/ft2) and temperature gradients as high as950 °F/ft. The resulting Knudsen numbers varied fromzero to approximately 10.
The dynamic frequency response tests demonstratedthe accuracy of the polytropic energy analysis,fundamental solution, and recursive formulation fortemperature gradients. Increasing temperature gradientsresulted in an increase in the phase delay of the responseand a shift of the spectral harmonics to higher frequencyand lower magnitude. This effect is verified extremelywell by the data. Up to the approximately 2000-Hzlimits of the data, the frequency response is predicted towithin the noise limits of the basic measurement.
The steady-state response tests verified the slip flowassumptions used in deriving the boundary valueequations and established an upper boundary on theapplicability of the model. Model comparisons are
excellent for Knudsen numbers up to around 0.6. Forvalues of , free-molecule effects begin todominate the flow, and the model analyses are no longervalid.
The model represents a fundamental contribution tothe understanding of flow behavior at the limits of thecontinuum flow regime. The model allowsinstrumentation designers to evaluate the responses ofpneumatic systems over a wide range of flow conditionsin a general and unified way without having to resort toad hoc or special case models.
APPENDIX ADEVELOPMENT OF BOUNDARY
VALUE EQUATIONS
This appendix develops the boundary value equationsfor the mathematical model. The basic strategy is tolinearize the fundamental equations of energy,continuity, and momentum by assuming small inputperturbations. The partial differential equations arereduced to ordinary differential equations using theFourier transform, and the boundary value equations aredeveloped in the frequency domain. To account for slipand rarefied flow effects, a slip flow boundary conditionis allowed at the tubing walls. The resulting equationsare averaged across the cross-section of the tube to givea one-dimensional model. For small tube diameters, noradial pressure gradients exist; therefore, little loss ingenerality occurs. The energy equation is decoupledfrom the equations of momentum and continuity byassuming the wave expansion in the tube to bepolytropic (ref. 12).
Coordinate Definitions andBasic Assumptions
The sensor configuration is modeled as a straightcylindrical tube with the internal volume of the pressuretransducer attached to its downstream end (fig. A-1).The total tube length is L. A longitudinal coordinate, x,is defined from the upstream (port) end of the tube, anda radial coordinate, r, is defined starting at the center ofthe tube. At each longitudinal station, the tube hasradius, R, not necessarily a constant for eachlongitudinal station. The density and velocitydistributions, , vary as a function oflongitudinal distance down the tube, radial distance
κ 0.6>
ρ x r t, ,( ) u x r t, ,( ),
11
from the center of the tube, and function of time. Theparameter U is the longitudinal velocity. Pressurevariations at the surface propagate as longitudinal wavesthrough the connective tubing to the transducer. Thewave propagation is damped by frictional attenuationalong the walls of the tubing. When the wave reachesthe downstream end of the tubing, it reflects back up thetube and may either damp or amplify incoming pressurewaves.
For small tubes, flow occurs only in the longitudinaldirection. Assuming that the system is initially at restand input disturbances are small, the second-order termsare neglected. In addition,
(A-1)
The temperature distribution in the tube is assumed tobe forced by heat transfer from or to outside sources andsinks. The wall temperature profile, , is assumedto be prescribed and known a priori. To simplify theanalysis at any given longitudinal station, thetemperature gradient is assumed to be constant. Thus,
(A-2)
Later, because the problem will be cast as a finite-element solution with a series of piecewise longitudinaltemperature variations, this assumption is not toorestrictive.
Unlike continuum flow conditions where fluidelements stick to the tubing wall and the classical no-slip boundary condition holds, for rarefied flowconditions, the fluid velocity at the wall is not zero.Large temperature gradients can result in the so-calledmolecular creep effect. This effect is primarily amolecular phenomenon where gas molecules adjacent tothe tube wall creep from the cold end of the tube to thehot end. Furthermore, under low-pressure conditions,such as those experienced at very high altitudes, tubeflow can become so rarefied that the fluid slips at thetube wall. Modification of this boundary conditionmakes the rarefied flow problem unique.
Energy Analysis
Based on the assumptions stated in the previoussection, the energy balance is (ref. 6)
(A-3)
where is thermal conductivity of the fluid, is thespecific heat at constant pressure, is the nominaldensity, T is the local temperature, and P is the localpressure. Solving for the time derivative of temperatureand taking the Fourier transform of equation (A-3),
(A-4)
where the is the dynamic viscosity, and isthe Prantl number.
Defining the shear wave number,
and a nondimensional
longitudinal coordinate where is the
bulk (radially averaged) viscosity of the fluid. Equation
(A-4) becomes
(A-5)
Equation (A-5) is a form of Bessel equation of orderzero (ref. 13) and has a general solution of the form
(A-6)
where is Bessel function of the zeroth order. Theparameter is evaluated using the boundarycondition at the wall, . Thus,
(A-7)
Equation (A-7) is now written as a one-dimensionalradially averaged
∂P∂r------ U
∂T∂x------ U
∂P∂x------
∂U∂x-------
2
… 0≈,,,
Tw x( )
∂2T
∂x2
--------- 0≈
ρ0Cp∂T∂t------ η ∂2
T
∂r2
---------1r---
∂T∂r------+ ∂P
∂t------+=
η Cpρ0
T µ
jωρ0 Pr-------------------- ∂2
T
∂r2
---------1r---
∂T∂r------+– P
ρ0Cp-------------=
µ Pr
µCp
η----------=
α j3 2⁄ ω ρ0 R
2( ) µ0⁄≡z α r
R--- Pr= µ0
∂2T
∂z2
---------1z---
∂T∂z------ T+ + P
ρ0Cp-------------=
T x r,( ) M x( ) J0 Pr α rR--- P
ρ0Cp-------------+=
J0M x( )
T x R,( ) Tw x( )=
T x r,( ) Tw x( )
J0 Pr α rR---
J0 Pr α[ ] ---------------------------------
1J0 Pr α
rR---
J0 Pr α[ ] ---------------------------------–
+ P x( )ρ0Cp-------------
=
12
(A-8)
where is the second-order Bessel function. Becauseno radial pressure gradients exist,
(A-9)
where is the universal gas constant, and is theratio of specific heats. Equation (A-8) is approximatedas
(A-10)
Polytropic Analysis
To decouple the energy equation from the equationsof momentum and continuity, density and temperatureare written in terms of pressure by assuming that thewave expansion process in the tube is polytropic(ref. 12).
(A-11)
Differentiating equation (A-11) with respect todensity
(A-12)
and temperature
(A-13)
where c is the local sonic velocity. Differentiatingequation (A-10) with respect to temperature gives
(A-14)
Comparison of equation (A-14) with equation (A-13)gives
(A-15)
Equation (A-15) is the same expression as derived byBergh using a different approach (ref. 6). Equations(A-15) and (A-10) and the equation of state for an idealgas are used to replace the energy equation throughoutthe remainder of this analysis.
Momentum Analysis
The Navier-Stokes momentum equation expressed incylindrical coordinates is (ref. 11)
(A-16)
Using the equation of continuity for tube flow,linearized for small disturbances
(A-17)
and the polytropic equation (A-13),
(A-18)
to eliminate , and taking the Fourier transform
(A-19)
Tavg x( )T x r,( )r rd θd
0
R
∫0
2π
∫πR
2------------------------------------------------
Tw= x( ) 1J2 Pr α[ ]
J0 Pr α[ ]--------------------------+
P x( )ρ0Cp-------------
J2 Pr α[ ]
J0 Pr α[ ]--------------------------–
=
J2
P x( )ρ0Cp-------------
RgTw
Cp-------------≈ γ 1–
γ----------- Tw=
Rg γ
Tavg x( ) Tw x( ) 11γ---
J2 Pr α
J0 Pr α
--------------------------+
=
P K ρξK' T
ξ ξ 1–( )⁄= =
∂P∂ρ------ ξ
Pρ--- Rg T
ξγ-- c
2= = =
∂P∂T------
ξξ 1–---------
PT---=
∂P∂T------
P x( )T x( )-----------
ρ0CpT x( )P x( )
------------------------- J0 Pr α[ ]
J2 Pr α[ ]--------------------------
–=
γ
γ 1–-----------
J0 Pr α[ ]
J2 Pr α[ ]--------------------------
P x( )T x( )-----------–≈
ξ 1
1γ 1–
γ-----------
J2 Pr α
J0 Pr α
--------------------------+
-----------------------------------------------------------=
ρ0 ∂U x r,( )
∂t--------------------- ∂P
∂x------
µ =1r---
∂∂r----- r
∂U x r,( )∂r
--------------------- 4
3---
∂2U
∂x2
----------+
+
∂ρ∂t------ ρ0
∂U∂x-------+ 0=
∂P∂t------
∂P∂ρ------
∂ρ∂t------
ξγ-- c
2 ρ0 ∂U∂x-------–= =
∂2U ∂x
2⁄
jω ρ0 U x r,( ) ∂P∂x------+
µ 1r---
∂∂r----- r
∂Ux r,∂r
----------------
43---
jωρ0------
∂∂x------
P
ξ γ⁄ c2
---------------
–=
13
Collecting terms gives
(A-20)
In addition,
(A-21)
where is the mean steady-state pressure at the localtemperature. Equation (A-20) becomes
(A-22)
where is a rarefied flow correction
term and for air is very close to unity except for rarefiedflow conditions. Using a solution method similar to theenergy analysis performed earlier, equation (A-22) isintegrated with respect to the radial coordinate to give(ref. 6)
(A-23)
The constant of integration, , is solved for usingthe boundary condition at the wall. For slip flowconditions, the fluid velocity at the wall can bedecomposed into two parts: slip velocity andthermomolecular creep velocity. Slip velocity isproportional to the shear stress at the wall. The creepflow is proportional to the longitudinal temperaturegradient at the wall. For laminar flow, the wall boundarycondition is (ref. 1)
(A-24)
The parameter, , is referred to as the slip distance.For a given material, this distance can be determined byreferring to empirical charts. The parameter is the
local steady-state bulk viscosity in the tubing. Applyingequation (A-23) to equation (A-22), solving for theparameter , and simplifying the solution for slipflow can be written as
(A-25)
where is the first-order Bessel function. Averagingover the cross-section of the tube gives
(A-26)
Because no radial pressure gradients occur, thepolytropic process equation is used to give
(A-27)
In addition,
(A-28)
U x r,( ) µjω ρ0--------------
1r---
∂∂r----- r
∂U x r,( )∂r
--------------------- –
1∂
jω ρ0∂x-------------------- 1
43---
µρ0-----
jω
ξ γ⁄( )c2
--------------------
+
P–=
ρ0 ξ γ⁄( ) c2 ξ ρ0 Rg T ξ P0≈=
P0
U x r,( ) µjω ρ0--------------
1r---
∂∂r----- r
∂U x r,( )∂r
--------------------- –
κ p∂P
jω ρ0∂x-------------------- –=
κ p 143---
jω µξ P0-----------+
=
U x r,( ) N x( ) J0 α rR---
κ p∂P∂x------
jωρ0-------------–=
N x( )
U x R,( ) Uslip Ucreep+=
ϑ ∂U∂r-------–
34---
µ0Rg
P0------------ ∂Tw ∂x⁄+≈
ϑ
µ0
N x( )
U x r,( )J0 α r
R---
J0 α rR--- ϑ
αR--- J1 α r
R---–
-------------------------------------------------------- 1– κ p
∂P∂x------
jω ρ0--------------=
J0 α rR---
J0 α rR--- ϑ
αR--- J1 α r
R---–
-------------------------------------------------------- Ucreep+
J1
Uavg x( ) 1
πR2
---------- U x r,( )r r d θd0
R
∫0
2π
∫J2 α[ ] ϑ
αR--- J1 α[ ]+
J0 α[ ] ϑ –αR--- J1 α[ ]
---------------------------------------------------- κ p
∂P∂x------
jωρ0---------------
=
=
3µRg
4P0-------------+
2α--- J1 α[ ]
J0 α[ ] ϑ– αR--- J1 α[ ]
----------------------------------------------- ∂Tw
∂x----------
∂Tw
∂x----------
∂P∂x------
∂P∂T-------------
ξ 1–ξ
----------- TP---
∂P∂x------= =
Uavg
κ p J2 α[ ] ϑ αR---J1 α[ ]+
jω34---
µ0
P0------
ξ 1–ξ
----------- 2α---J1 α[ ]+
J0 α[ ] ϑ αR--- J1 α[ ]–
-------------------------------------------------------------------------------------------------------------------------≈
∂P∂x------
jω ρ0--------------×
14
Defining a propagation factor
(A-29)
the momentum equation for slip flow conditions canfinally be written as
(A-30)
Continuity Analysis
Based on assumptions in eq. (A-1), the radiallyaveraged continuity equation is
(A-31)
In addition, the Fourier transform of equation(A-31) is
(A-32)
Differentiating equation (A-32) with respect to x, andsubstituting into equation (A-31)
(A-33)
Equation (A-33) is the final form of the wave equationwhich describes the slip flow propagation of pressurewaves in the tube.
Evaluation of the DownstreamBoundary Condition
At the downstream end of the tube where the pressurewave exits the pressure tubing and enters the transducervolume, the equation of momentum (eq. (A-32)) stillholds, but the equation of continuity must be modified.Here, the integral form of the equation is used and
(A-34)
where V is the entrapped transducer volume, and isthe cross-sectional area of the tube at the exit to thetransducer. Substituting equation (A-34) into equation(A-32), the downstream boundary condition becomes
(A-35)
Equations (A-27), (A-28), (A-33), and (A-35) are thecollected boundary value equations.
APPENDIX BSOLUTION OF THE BOUNDARY
VALUE EQUATIONS
As derived in appendix A, the pressure wavepropagation equations are
(B-1)
The downstream boundary condition is
(B-2)
Pressure at the upstream boundary is prescribed.Because several parameters of the boundary valueequations vary as a function of the longitudinaltemperature distribution in the tube, the equationsgenerally cannot be integrated outright. Conceptually,the set of boundary value equations can be integrated byformulating the problems as a finite difference solution.Unfortunately, the wave equation is hyperbolic, and thedownstream boundary condition is parabolic (ref. 12).This mis-match of equation types makes the problem ill-conditioned for finite difference methods. Depending onthe ratio of the time step to the distance step, variousdegrees of artificial damping will be introduced into thesystem by a finite difference formulation. This artificialdamping makes extracting the true physics fromnumerical artifacts extremely difficult.
Γ p
γξ--
J0 α[ ] ϑ αR--- J1 α[ ]–
κ p J2 α[ ] ϑ αR--- J1 α[ ]+
jω34---
µ0
P0------
ξ 1–ξ
----------- 2α---J1 α[ ]+
---------------------------------------------------------------------------------------------------------------------------≡
∂P∂x------ jω
ξγ-- Γ p
2 ρ0 Uavg= x( )
∂ρ∂t------
∂P∂t------
∂P∂ρ------
------------γξ--
∂P∂t------
1
c2
----- ρ0 ∂Uavg
∂x---------------
–= = =
jωP ξγ--c
2 ρ0
∂Uavg
∂x---------------
– 0= =
∂2P x( )
∂x2
----------------- ωΓ p
c------
2
P x( )=
Uavg L( ) jωρVρ0 Ac--------------
jωPVρ0 Ac--------------
ξγ--c
2--------------= =
Ac
∂P∂x------
x L= ω2Γ p
2
c2
------------- VAc------ PL–=
∂2P x( )
∂x2
----------------- ωΓ p
c------
2
P x( )=
∂P∂x
x L=
------------------- ω2Γ p
2
c2
------------- VAc------ PL–=
15
Instead, a better approach is to integrate the boundaryvalue equations with respect to x, assuming that the gasproperties remain constant along the region ofintegration. This longitudinally averaged model can besolved in closed form in the frequency domain to give afundamental solution. Using this fundamental solutionas a building block, the problem can be solvedrecursively starting at the transducer end and workingtoward the surface end of the tube. If the properties ofthe flow are re-evaluated at each new node, then onemay effectively allow for the construction of a solutionin which the flow properties are arbitrarily variable as afunction of x. This finite-element approach is an integralmethod that is not subject to the numerical problemsencountered with the finite difference methods.
Fundamental Solution ofthe Boundary Value Equations
Across the tube, assume constant flow properties.Integrating equation (B-1) with respect to x gives
(B-3)
Applying the upstream and downstream boundaryconditions to solve for and , substituting theresults into equation (B-3), and simplifying gives
(B-4)
Evaluating equation (B-4) at gives the end-to-end solution:
(B-5)
Equation (B-5) is a frequency response model wherethe complex spectra are given as a function of the sensorgeometry, the frequency of the input sinusoid, and thepropagation factor, Molecular effects are allembedded in If molecular effects are ignored,equations (B-4) and (B-5) are identical to thefundamental solutions developed by Bergh, et al.,(refs. 6, 7).
Recursive Solution of theBoundary Value Equations
The fundamental solution is limited to applicationswhere the temperature gradients are small, and the tuberadius is constant. However by using equations (B-4)and (B-5) as building blocks, solutions allowinglongitudinal variation of the fluid properties andcomplex tube geometries can be constructed. Thesolution is recursive and moves from the downstreamboundary to the upstream boundary. The solution isperformed assuming n solution elements. Theseelements are not necessarily evenly spaced. Thejunctions of elements are referred to as nodes. Withineach element, the fluid properties are assumed to beconstant, but properties between nodes are allowed tovary. At each node, the equation of continuity issatisfied, giving a new downstream condition. Forgenerality, the tube radius is allowed to vary, and avolume is assumed to be entrapped at each node. Theseentrapped volumes can be used to model the effects oftube joints, fittings, or other devices, such as water traps.
First, consider a two-node system with node n beingthe transducer node and node n-1 being the adjacentupstream node. The configuration being analyzed isdepicted in figure A-1. At node n-1, the integral form ofthe continuity equation is
(B-6)
Substituting the downstream boundary condition(eq. (B-2)) for yields
P x ω,( ) Aωe ωΓp xc--
Bωe ωΓp xc--–
+
=
Aω Bω
P x ω,( )
P0 ω( )ωΓp
x L–c
------------ V αAcc
------------ ωΓp x L–
c------------sinh–cosh
ωΓp Lc---
VωΓp
Acc--------------- ωΓp
Lc---sinh+cosh
-----------------------------------------------------------------------------------------------------------=
x L=
PL ω( )
P0 ω( ) 1
ωΓp Lc---
VωΓp
Acc--------------- ωΓp
Lc---sinh+cosh
---------------------------------------------------------------------------------------------=
Γ p.Γ p.
Un 1–jωρ0------
Vn 1–
Acn 1–
------------- Pn 1–
ξ γ⁄ cn 1–2
------------------------
=
Acn
Acn 1–
------------- Pξγ-- c
x ω2, ⁄ x d
LN 1–
LN
∫+
Vn
Acn 1–
------------- Pn
ξ γ⁄( ) cn2
----------------------+
Un 1–
16
(B-7)
To evaluate the integral in equation (B-7), thefundamental solution (eq. (B-4)) is used. Thefundamental solution is valid from node n-1 to node nbecause the downstream boundary condition is identicalto that of the fundamental solution. Performing theintegration, simplifying, and collecting terms equation(B-7) reduces to
(B-8)
where is the effective volume parameter whichaccounts for the impedance of the downstream tube andvolume as well as the volume at node n-1.
(B-9)
The form of the new boundary condition given byequation (B-8) is identical to the original boundarycondition at node n. By induction on the fundamentalsolution (at node n), the solution at node n-1 is
(B-10)
To establish generality, the process must be repeatedan additional time. Repeating the process at node n-2,the downstream boundary condition is
(B-11)
However, from the solution at node n-1,
(B-12)
∂P n 1 ω,–( )∂x
------------------------------- Vn 1–
Acn 1–
------------- ω2
Γ pn 1–
2
cn 1–2
----------------------- Pn 1–
–=
Acn
Vn 1–-------------+
cn 1–2
cn2
------------ Ln 1–
Ln
∫ P dx
cn 1–2
cn2
------------ Vn
Vn 1–------------- Pn
+
∂P n 1–( )∂x
------------------------ Ven 1–
Acn 1–
------------- ω2
Γ pn 1–
2
cn 1–2
----------------------- pn 1––=
Ve
Ven 1–Vn 1–=
cn 1–2
Vn
cn2
-------------------+ ωΓpn Ln
cn------cosh
1
ω Vn
Acn
-------- Γ pn
cn--------
------------------------- ωΓpn Ln
cn------sinh+
ωΓpn Ln
cn------cosh
÷
ω Vn
Acn
-------- Γ pn
cn-------- ωΓpn
Ln
cn------
sinh+
Pn 1– ω( ) Pn 2– ω( ) =
ωΓpn 1– Ln 1–
cn 1–-------------cosh
÷
ω Ven 1–
Acn 1–
------------- Γ pn 1–
cn 1–-------------- ωΓpn 1–
Ln 1–
cn 1–-------------
sinh+
∂Pn 2–
∂x---------------- ω
Vn 2– Γ pn 2–
Acn 2–cn 2–
2----------------------------- Pn 2–
–=
Acn 1–cn 2–
2
Vn 2– cn 1–2
-------------------------- P xdLn 2–
Ln 1–
∫+
cn 2–2
Vn 1–
cn 1–2
Vn 2–
-------------------------- Pn 1–
+
cn 1–2
Acn
cn2Vn 1–
--------------------- P xdLn 1–
Ln
∫+
Vn
Vn 1–-------------
cn 1–2
cn2
------------ Pn
+
Pn 1–
Acn
cn 1–2
Vn 1– cn2
------------------------- P xdLn 1–
Ln
∫+
Vn
Vn 1–-------------+
cn 1–2
cn2
------------ Pn
Ven 1–
Vn 1–------------- Pn 1–=
17
In addition, equation (B-12) becomes the newboundary condition at node n-2.
(B-13)
This form of equation (B-13) is identical to theboundary condition at node n-1 (eq. (B-7)). Again byinduction on the solution at node n-1, the solution atnode n-2 can be written as
(B-14)
where the effective volume at node n-2 is
(B-15)
General Solution UsingRecursive Formulation
The general end-to-end frequency response is givenas the complex product of the frequency responses at theindividual nodes. By induction on the previoussolutions, the solution at the ith node is
(B-16)
where the effective volume is
(B-17)
with . In addition,
(B-18)
∂Pn 2–
∂x---------------- ω
Vn 2– Γ pn 2–
Acn 2– cn 2–
2------------------------------ Pn 2–
–=
Acn 1– cn 2–
2
Vn 2– cn 1–2
------------------------------ + P xdLn 2–
Ln 1–
∫
cn 2–2
Ven 1–
cn 1–2
Vn 2–
----------------------------+ Pn 1–
Pn 2– ω( ) Pn 3– ω( ) =
ωΓpn 2– Ln 2–
cn 2–-------------cosh
÷
ω Ven 2–
Acn 2–
------------- Γ pn 2–
cn 2–-------------- ωΓpn 2–
Ln 2–
cn 2–-------------
sinh+
Ven 2–Vn 2–=
cn 1–2
Ven 1–
cn 1–2
--------------------------+ ωΓpn 1– Ln 1–
cn 1–-------------cosh
1
ω Ven 1–
Acn 1–
------------- Γ pn 1–
cn 1–--------------
------------------------------------ ωΓpn 1– Ln 1–
cn 1–-------------sinh+
ωΓpn 1– Ln 1–
cn 1–-------------cosh
÷
ω Ven 1–
Acn 1–
------------- Γ pn 1–
cn 1–-------------- ωΓpn 1–
Ln 1–
cn 1–-------------
sinh+
Pi ω( )
Pi 1– ω( )
ωΓpi Li
ci----- ω
Vei
Aci
------- Γ pi
ci------- ωΓpi
Li
ci-----sinh+cosh
----------------------------------------------------------------------------------------------------------=
VeiV i=
ci2Vei 1+
ci2
------------------+ ωΓpi 1+ Li 1+
ci 1+------------cosh
1
ω Vei 1+
Aci 1+
------------- Γ pi 1+
ci 1+-------------
----------------------------------- ωΓpi 1+ Li 1+
ci 1+------------sinh+
ωΓpi 1+ Li 1+
ci 1+------------cosh
÷
ω Vei 1+
Aci 1+
------------- Γ pi 1+
ci 1+------------- ωΓpi 1+
Li 1+
ci 1+------------
sinh+
VenVn=
PL ω( )P0 ω( )----------------
P1 ω( )P0 ω( )---------------=
P2 ω( )P1 ω( )--------------- …
Pn 1– ω( )Pn 2– ω( )----------------------
Pn ω( )Pn 1– ω( )----------------------
1
ωΓpi Li
ci----- ω
Vei
Aci
------- Γ pi
ci------- ωΓpi
Li
ci-----sinh+cosh
---------------------------------------------------------------------------------------------------------
i 1=
n
∏=
18
Equations (B-17) and (B-18) represent a finite-element solution of the boundary value problem. Theboundary value equations have been analyticallyintegrated at each element based on simplifyingassumptions. The method is not subject to the numericalproblems encountered with the finite differencemethods. The solution assumes that a tube surfacetemperature profile is prescribed at each node, andthe fluid properties, namely the temperature, sonicvelocity, and dynamic viscosity, are evaluated as afunction of the prescribed temperature using the energyequation (A-10).
APPENDIX C
STEADY-STATE RESPONSE OF
TUBE RESPONSE MODEL
FOR RAREFIED FLOW
CONDITIONS
If the general solution (eq. (B-18)) is evaluated at lowfrequency with large longitudinal temperature gradientsand rarefied flow conditions, the gain does notapproach 1 as it does for continuum flow conditions.Instead, the hot end of the tube has a higher pressuremagnitude than the cold end. This equilibrium pressuregradient is a well-known result. The equilibriumbehavior of the model for rarefied flow conditions isbest understood by looking at the momentum equationfor a cross-section of the pressure tubing.
Evaluating the steady-state behavior of equation(A-28) using the final value theorem yields (ref. 16)
(C-1)
Using the series expansion form for Bessel functionsyields
(C-2)
Then,
(C-3)
In addition,
(C-4)
but for equilibrium flow conditions, there must be no netflow across any cross-section of the tube, and .As a result,
(C-5)
but . Using the polytropic process
energy equation (eq. (A-13)) after some simplificationyields
(C-6)
If the ratio of slip distance to mean free path isdefined as , then
(C-7)
Equation (C-6) becomes
(C-8)
Ussjω
lim ω 0⇒-------------------------=
κ p J2 α[ ] ϑ αR--- J1 α[ ] jω3
4---
µ0
P0------
ξ 1–ξ
--------- 2α--- J1 α[ ]+ +
J0α ϑ –αR--- J1α
-----------------------------------------------------------------------------------------------------------------------
∂P∂x------
jωρ0------------×
Jn α[ ] 1–( )k
k! n k+( )!-------------------------
α2---
n 2k+
k 0=
∞
∑=
J0 α[ ]limω 0⇒------------------------ 1
J1 α[ ]limω 0⇒------------------------
α2---,=,=
J2 α[ ]limω 0⇒------------------------
α2
8------ κ p, 1= =
Uss1ρ0----- α2
8------
ϑR---
α2
2------ jω
34---
µ0
P0------
ξ 1–ξ
-----------+ +
∂P∂x------=
Uss 0=
α2
8------ 1 4
ϑR---+
∂P∂x------ jω
34---
µ0
P0------
ξ 1 ∂P– ξ ∂x---------------------–=
α2 jω
ρ0 R2
µ0--------------–=
∂P∂x------
P-------
6
1 4ϑR---+
---------------------
µ02
ρ0 R2 P0
-----------------------
∂T∂x------
T-------=
6
1 4ϑR---+
---------------------=
µ02 Rg T0
R2 P0
2-----------------------
∂T∂x------
T-------
ε ϑ λ⁄=
κ2 πRg µ2
R2
------ T
P2
------ ϑR---,≈ ε
λR--- ε κ= =
∂P∂x------
P-------
6κ02
π 1 4 ε κ0+( )---------------------------------
∂T∂x------
T-------≈
19
Clearly, based on equation (C-8), the normalizedpressure gradient is proportional to the normalizedtemperature gradient and mean properties of the flowgiven by . This equation is used to evaluate the rangeof Knudsen numbers for which the rarefied flow modelis valid.
REFERENCES
1Kennard, Earle H., Kinetic Theory of Gases,McGraw-Hill, New York, 1938, pp. 311–337.
2Iberall, Arthur S., Attenuation of OscillatoryPressures in Instrument Lines, U.S. National Bureau ofStandards Report RP2115, vol. 45, July 1950.
3Lamb, J.P., Jr., The Influence of Geometry ParametersUpon Lag Error in Airborne Pressure MeasurementSystems, WADC TR 57-351, Wright-Patterson AFB,Ohio, July 1957.
4Schuder, C.B. and Binder, R.C., “The Response ofPneumatic Transmission Lines to Step Inputs,”Transactions of the American Society of MechanicalEngineers, Dec. 1959.
5Hougen, J.O., Martin, O.R., and Walsh, R.A.,“Dynamics of Pneumatic Transmission Lines,” Journalof Control Engineering, March 1960.
6Bergh, H. and Tijdeman, H., Theoretical andExperimental Results for the Dynamic Response ofPressure Measuring Systems, NLR-TR F.238, NationalAero- and Astronautical Research, Amsterdam,Jan. 1965.
7Tijdeman, H. and Bergh, H., The Influence of theMain Flow on the Transfer Function of Tube-Transducer Systems Used for Unsteady PressureMeasurements, NLR-MP 72023, National Aero- andAstronautical Research, Amsterdam, 1972.
8Tijdeman, H., Investigations of the Transonic FlowAround Oscillating Airfoils, NLR-TR 77090, NationalAero- and Astronautical Research, Amsterdam, 1977.
9Parrot, T. and Zorumski, W., “Sound TransmissionThrough a High-Temperature Acoustic Probe Tube,”AIAA 90-3991, Oct. 1990.
10Knudsen, von Martin, “Eine Revision derGleichgewichtsbedingung der Gase: ThermischeMolekularströmung,” Annalen der Physik, vol. 31,Nov. 1910, pp. 205–229.
11Stephens, R.W.B. and Bate, A.E., Acoustics andVibrational Physics, St. Martin Press, New York, 1966.
12Freiberger, W.F., ed., The International Dictionaryof Applied Mathematics, D. Van Nostrand Co., Inc.,Princeton, 1960, pp. 17.
13Lennart, Rade and Westergren, Bertil, BetaMathematics Handbook, CRC Press, Boca Raton, 1992.
14Bendat, Julius S. and Piersol, Allan G., RandomData: Analysis and Measurement Procedures, Wiley &Sons, New York, 1971.
15Doebelin, Ernest O., Measurement Systems:Application and Design, 2nd ed., McGraw-Hill, NewYork, 1983, pp. 404–456.
16Franklin, Gene F. and Powell, J. David,Digital Control of Dynamic Systems, Addison Wesley,Reading, 1980.
κ0
20
TABLES
Table 1. Frequency response test matrix.
Diameter, Length, Thermocouple readings along tube, °F
Run in. in. 1 2 3 4 5 6
1 0.066 12 74.0 74.0 75.0 74.0 77.0 84.2
2 0.066 12 74.8 225.1 338.0 344.0 218.0 85.1
3 0.066 12 77.8 309.0 496.0 503.0 312.0 89.1
4 0.066 12 79.4 381.0 636.0 644.0 385.0 90.1
5 0.066 12 80.4 383.0 639.0 647.0 387.0 92.9
6 0.066 12 78.0 311.0 501.0 508.0 315.0 92.4
7 0.066 12 77.0 228.0 340.0 348.0 221.0 88.9
8 0.066 12 77.0 77.2 77.4 78.0 77.4 88.4
9 0.033 12 74.0 74.0 75.0 74.0 77.0 84.2
10 0.033 12 77.0 233.3 354.0 360.9 230.0 91.8
11 0.033 12 79.8 316.3 501.2 511.8 312.0 105.3
12 0.033 12 81.6 392.6 641.5 652.3 389.8 113.3
13 0.033 12 81.7 392.4 641.4 653.1 390.0 113.5
14 0.033 12 79.8 317.0 502.3 512.2 312.4 105.4
15 0.033 12 75.0 231.6 352.0 359.9 230.0 91.5
16 0.033 12 77.0 77.0 77.3 77.2 77.0 88.2
Table 2. Rarefied flow condition test matrix.
Diameter, Tube temperatures, °F Pressure, psf
in. Hot Cold Minimum Maximum
0.014 499.700 75.600 0.28 1952.2
0.014 950.700 75.600 0.56 1953.3
0.033 499.700 75.600 1.68 1953.3
0.066 433.000 81.000 1.87 1951.2
0.066 953.300 83.900 3.50 1949.1
0.080 949.800 84.200 1.69 1951.2
0.092 431.800 82.100 1.99 1949.6
0.092 501.200 76.800 2.80 1951.0
21
FIGURES
Figure 1. Knudsen number ranges for continuum, slip, and molecular flow regimes.
(a) Sensor geometry for heating tests.
Figure 2. Experimental apparatus used for frequency response tests.
Continuumregime ~
Navier-Stokes
Free-moleculeregime ~
Statisticalmechanics
.01 1 10Knudsen number
.10 ∞
Slip-flowregime
Navier-Stokeswith slip boundary
conditions
Conventionalaeronautics
Hypersonicflight
Orbital andsuborbital
Validity range of dynamic model
960019
12 in.
4.2 in.
0.5 in.
1.30 in. 3.9 in.
0.066 in.or
0.033 in.
Test microphone
Reference microphone (flush mounted)
Aluminum test plate
1.30 in.
Delrin spacer
Voltage output tosignal amplifier
Voltage output tosignal amplifier
Aluminum rodPort
Volume = 0.0026 in3
TC = Type T thermocouples
Brass tubing
3.9 in.
1.30 in. 1.30 in.
TCTC TC TC TC TC
22
(b) Equipment layout.
(c) Speaker mounted in sound chamber and the heater apparatus and test plate.
Figure 2. Concluded.
Heater with test thermocouples
Thermocouple panel with rotary switch
Reference microphone
Amplifiers and signal condition units for microphones
Microcomputer for data logging
Sound chamber
Temperature controller
Test microphone
Barometer for ambient conditions
Stereo amplifier, power supply, and voltage attenuator
960021
Reference microphoneTest thermocouples
Test plate
Test microphone
Fiberglass heating tape
Brass tubing
960022
23
(a) Time history.
(b) Spectrum.
Figure 3. Typical phase-modulated output waveform used for frequency response tests.
– 10
– 5
0
5
10
15
.20.15.10Time, sec
.050
Output,volts
960023
– 150
– 100
– 50
0
50
10,0001,00010010 Frequency, Hz
10
Magnitude,dB
960024
24
(a) Equipment layout.
Figure 4. Apparatus used for the static response and rarefied flow tests.
960025
Vernier manometerPressure valve
Vacuum pump
Cold end of heated tube
Vacuum oven
Pressure feedlines
Mcleod gauge
Thermocouple panel
25
(b) Front view of vacuum oven.
(c) Rear view showing pressure fittings.
Figure 4. Concluded.
Temperature control
Hot end of tube Thermocouple
Dial vacuum gauge
Vacuum pump
960026
Hermetically sealed plugs
Pressure valve
Thermocouple
Cold end of heated tube
960027
26
(a) Tube diameter, 0.066 in.
(b) Tube diameter, 0.033 in.
Figure 5. Idealized temperature profiles for frequency response tests.
14
960028
121086420
Temperature,°F
100
200
300
400
500
600
700
T1
T2
T3
T4
T5
T6
Longitudinal coordinate, in.
A
B
C
D
14
960029
121086420
Temperature,°F
100
200
300
400
500
600
700
T1
T2
T3
T4
T5
T6
Longitudinal coordinate, in.
A
B
C
D
27
Figure 6. The effects of grid density on frequency response computation (d = 0.066 in. and temperature profile D).
– 6
– 4
– 2
0
2
4
6
8
10
– 8
12
20 50 100 200Frequency, Hz
500 1000 2000 500010 5000
200
150
100
50
0
– 50
– 100
– 150
– 200
960030
Magnitude,dB
Phaseangle,deg
Elements
5102050
100
28
(a) L = 12 in. and d = 0.066-in. tube geometry.
Figure 7. Theoretical effects of temperature gradients.
960031
200
150
100
50
0
– 50
– 100
– 150
– 20010 20 50 100 200 500 1000 2000 5000
Frequency, Hz
Phaseangle,deg
15
10
5
0
– 5
– 10
A (Ambient)
B (Tmax = 350 °F)
C (Tmax = 500 °F)
D (Tmax = 650 °F)
Temperatureprofile
Magnitude,dB
29
(b) L = 12 in. and d = 0.033-in. tube geometry.
Figure 7. Concluded.
200
150
100
50
0
– 50
– 100
– 150
– 20010 20 50 100 200 500 1000 2000 5000
Frequency, Hz
Phaseangle,deg
10
5
0
– 5
– 15
– 25
– 20
– 10A (Ambient)B (Tmax = 350 °F)C (Tmax = 500 °F)D (Tmax = 650 °F)
Temperatureprofile
Magnitude,dB
30
(a) Temperature profile A (ambient conditions).
Figure 8. Comparison of model frequency response to experimental results for L = 12 in. and d = 0.066-in. tubinggeometry.
15
10
5
0
–5
–10
–15
–20
–25
–30
200
150
100
50
0
–50
–100
–150
–200
Phase angle, deg
Magnitude, dB
Laboratory data Model
960033
10 20 50 100 200
Frequency, Hz
500 1000 2000 5000
31
(b) Temperature profile B (Tmax = 350 °F).
Figure 8. Continued.
200
150
100
50
0
–50
–100
–150
–200
Phase angle, deg
15
10
5
0
–5
–10
–15
–20
–25
–30
Magnitude, dB
Laboratory data Model
960034
10 20 50 100 200
Frequency, Hz
500 1000 2000 5000
32
(c) Temperature profile C (Tmax = 500 °F)
Figure 8. Continued.
15
10
5
0
–5
–10
–15
–20
–25
–30
200
150
100
50
0
–50
–100
–150
–200
Phase angle, deg
Magnitude, dB
Laboratory data Model
960035
10 20 50 100 200
Frequency, Hz
500 1000 2000 5000
33
(d) Temperature profile D (Tmax = 650 °F).
Figure 8. Concluded.
10
5
0
–5
–10
–15
–20
–25
–30
200
150
100
50
0
–50
–100
–150
–200
Phase angle, deg
Magnitude, dB
960036
10 20 50 100 200
Frequency, Hz
500 1000 2000 5000
15 Laboratory data Model
34
(a) Temperature profile A (ambient conditions).
Figure 9. Comparison of model frequency response to experimental results (L = 12 in. and d = 0.033-in. tubinggeometry).
200
150
100
50
0
–50
–100
–150
–200
Phase angle,
deg
10
0
–10
–20
–30
–40
Magnitude, dB
Laboratory data Model
960037
10 20 50 100 200
Frequency, Hz
500 1000 2000 5000
20
35
(b) Temperature profile B (Tmax = 350 °F).
Figure 9. Continued.
200
150
100
50
0
–50
–100
–150
–200
Phase angle, deg
960038
10 20 50 100 200
Frequency, Hz
500 1000 2000 5000
Laboratory data Model
10
0
–10
–20
–30
–40
Magnitude, dB
20
36
(c) Temperature profile C (Tmax = 500 °F).
Figure 9. Continued.
200
150
100
50
0
–50
–100
–150
–200
Phase angle, deg
960039
10 20 50 100 200
Frequency, Hz
500 1000 2000 5000
Laboratory data Model10
0
–10
–20
–30
–40
Magnitude, dB
20
37
(d) Temperature profile D (Tmax = 650 °F).
Figure 9. Concluded.
200
150
100
50
0
–50
–100
–150
–200
Phase angle, deg
960040
10 20 50 100 200
Frequency, Hz
500 1000 2000 5000
Laboratory data Model10
0
–10
–20
–30
–40
Magnitude, dB
20
38
Figure 10. Comparison of steady-state model response to measured values of nondimensional steady responseparameter for rarefied flow conditions.
Figure A-1. Simple tubing geometry.
Figure A-2. Two-node geometry.
Free-molecule limit
Steady model response
.5
.4
.3
Ψ
.2
.1
0.01 .10
Average Knudsen number in tube960041
1 10
L
D VP0 P
LT
LT0 P(x, t)
T(x, t)
U(x, t)
Pn
Tn
VnDn
LnDn–1
Pn–1
Tn–1
Vn–1
Ln–1
39
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NSN 7540-01-280-5500 Standard Form 298 (Rev. 2-89)Prescribed by ANSI Std. Z39-18298-102
A Dynamic Response Model for Pressure Sensors in Continuum and HighKnudsen Number Flows with Large Temperature Gradients
WU 52-00-RR-00-000
Stephen A. Whitmore, Brian J. Petersen, and David D. Scott
NASA Dryden Flight Research CenterP.O. Box 273Edwards, California 93523-0273
H-2083
National Aeronautics and Space AdministrationWashington, DC 20546-0001 NASA TM-4728
This paper develops a dynamic model for pressure sensors in continuum and rarefied flows with longitudinaltemperature gradients. The model was developed from the unsteady Navier-Stokes momentum, energy, and continuityequations and was linearized using small perturbations. The energy equation was decoupled from momentum andcontinuity assuming a polytropic flow process. Rarefied flow conditions were accounted for using a slip flow boundarycondition at the tubing wall. The equations were radially averaged and solved assuming gas properties remain constantalong a small tubing element. This fundamental solution was used as a building block for arbitrary geometries where fluidproperties may also vary longitudinally in the tube. The problem was solved recursively starting at the transducer andworking upstream in the tube. Dynamic frequency response tests were performed for continuum flow conditions in thepresence of temperature gradients. These tests validated the recursive formulation of the model. Model steady-statebehavior was analyzed using the final value theorem. Tests were performed for rarefied flow conditions and compared tothe model steady-state response to evaluate the regime of applicability. Model comparisons were excellent for Knudsennumbers up to 0.6. Beyond this point, molecular affects caused model analyses to become inaccurate.
Hypersonic aerodynamics; Knudsen number; Pneumatic attenuation; Pressure sensing;Rarefied flow
A03
43
Unclassified Unclassified Unclassified Unlimited
January 1996 Technical Memorandum
Available from the NASA Center for AeroSpace Information, 800 Elkridge Landing Road, Linthicum Heights, MD 21090; (301)621-0390
Presented as AIAA 96-0563 at the 34th AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada,Jan. 15–18, 1996. Stephen A. Whitmore, Dryden Flight Research Center, Edwards, California; Brian J. Petersen, UCLA,Los Angeles, California; David D. Scott, Lawrence Livermore National Laboratories, Livermore, California.
Unclassified—UnlimitedSubject Category 02