Fluid velocity measurement by doppler shift of scattered light

ARCF-HEF

FLUID VLOCITY MEASUREMENT BY DOPPLER SHIFT OF

SCATTERED LIGHT

Submitted to the Fluid Dynamics Branch, Office of Naval Research, Washington, D.C., as a part of A Progress Report on Contract

Nonr 3963 (10)

October 1, 1968

Massachusetts Institute of Technoloay Williarfi B. Clarke

Department of Chemical Engineering July, 1968

Distribution of this

document is unlimited.

Supported by: ONR Contract Nonr-3963(10) and the Colonial Sugar Refining Co. Ltd.

Principal Investigators: Prof. F. W. Merrill Prof..K. A. Smith Prof. P. S. Virk

Lab.

v.

Scheepsbouwkuncie

Technische Hogeschool

Deift

INDEX

1. Introduction . . . . 1

2.. Resolution Requirements ...

2

3. Principle of Doppler Velocimetry . . 3

4. EquipinentRequired ... .

. ... .4

Light Source 5 ScatterincT Centres 5

c.LightDetector.

. .

... .

6 d. Other Components 5. Errors in Method . . . 11

6. Desc.tiption of Instrument,s Already Built . . . 15

7. Summary...-

... .

. 17

8. COnclusions 19

9. Appendices 20

Calculation of Intensity 20

Effect of Medium Change on Focussing 26

Broadening of signal due to Discrete Particles . . . . 29

1. Introduction

The standard instruments for measuring

velocitiesin

turbulent l.quid flow are impact tubes and hot film sensors. These have been used in the study of the Toms' Phenomenon

(see e.g. Virk1) hut are not entirely satisfactory. For

instance, impact tubes give accurate results only when they have a diameter iarger than about 1 mm. Hot film sensorS have unexplained flat teions in their calibration curves,

Yeh and Cummins2 in 1964 showed that fluid flow cOuld be measured by means of the Doppler shift in light scattered from the fluid. Since t1en other workers have developed instruments

(Foreman et al3'4'5, James, Seifert and Babcock6, Seifert,

7 8 . 9,10

Tullis and Morse, Berman and Santos ,Goldstein et al .)

One firm, Brown Engineering markets an instrument for

liquid

velocity measurement.

A Doppler velocimeter, using a laser as a light source, has some very useful characteriStics. It has very rapid

response, it is very sensitive (0.004 cm/sec is quoted in ref.2),. it has a linear response (output frecTuency vs. ve]ocity), it

does not disturb the flow, and the scattering volume can be made small (aoproachina the wavelength of the ljght used).

The Doppler Velocimeter appeared to he a suitable instrument for studying the

ToTn'

Phenomenon. This report describes the

velocj.meter and its design to study turbulent flow, and tatés the conclusions reached about its suitability.

-2-2. Resolution Requirements in Turbulertt Flow

If measurements are to be carried out in turbulent flow in the boundary layer, it is convenient to use UT

(=ftw/p)

as

a velocity scale and \/u as a length scale. (i) Liauids

Consider water flOwing in a 5 cm dia. pipe with. average velocities of 20 and 1000 cm/sec (Re = 1O4, and 5 x 1O5 resp.). Then the friction factor, f, equals 0.0079 and 0.0030,

and _UT = 1.26 and 38.7 cm/sec respectively. The length scales are 80 and 2.6 urn respectively.

For polymer soutions, there is another approach. Virk1 and others have shown that the value of ü at the onset of drag reduction can be correlated with polymer properties. For example the table below shows. the velocity and length scales

t.onset of drag reduction for poly(ethylene oxide) dissolved

in water.

A velocimeter used to study turbulence in liquids should have a spatial resolution of the order of 10 to 50 ii', preferably

below 20 urn.

-Polymer

-Turbulence scales at onset of drag reduction

Type M t (x106)

17elcity)

Length (iirn)

PEO Nb 0.092 12 4.5

PEO N750 0.63 8 10.5

PEO N3000. 0.76 7 12

Fluid flowing

in pipe

(ii) Gases

A similar calculation can be made for air flowing in a 25 cm dia. duct at average velocities of 60 and 3000 cm/sec

(Re = and 5 x 1O5 respectively). In this case the velocity

scales u. are 38 and 116 cm/sec, and the length scales are

400 and l3im respectively

The $patial resolution requirements for a gas are somewhat less strict-than for a liquid.

3. Principle of DQppler Velocimeter

The Doppler effect occurs when the detector of

electro-magnetic or SOund waves is roving relative tO the emitter of ---those waves. When this happens the frequency of the detecte4

wave differ$ from that of the einitted wave by an amount which is proportional to the radial component of the velocity.

The effect Occurs twice if waves are emitted from a

stationary sourc then reflected or scatteed by a moving object, and detected by. a stationary detector. This principle has been used -for a long time in Doppler radar (as used by police), and

in Sonar (sUbmarine c9etection by sound waves). It is also

possiL'le to use a light beam as the er-it.ted wave, ine. a gas

laser is a light source of discrete frecuencies. Hence the velocity Of a fluid can he i'easured if it contains suspended oárticles, or dissoive

macromolecules.-For lio'uid velocity measurements the basic equipment layout is shown in Fig. 1: (ignoring refraction effects at the

jnter-faces).

centre

4Lens

__Light source

-4-With this simple system, the velocity component measured is that which bisects the angle between the. incident beam and the scattered beam, and which lies in the plane of them. If

three Separate detectors are used, it is theoretically possible to simultaneously méausré all three velocity components.

If n (the. refractive index of the. liquid),A0 (the. source

wavelength, in vacuo), and 0 (the scàtterincy angle) are known, the frequency dif.fCrence,

D' between the emitted light and the

scattered light is related to the velocity

component

v by the

following equation: .

2 nv

=

Ao sin

(which is derived in ref. 2).

Typical values are

(kflz) = 4.06 v (cm/sec) for e = 100 = 32.9 v (cm/sec) for 0 = 900 when

= 0.6328 pm (helium neon gas laser)

and n = .1.474 (organic solvent Such as decalin)

Then, for liquid flow, typica frequency differences range from a few kHz to a few MHz.

Equipment Required

The 3 basic elements of a velocimeter are shown in Fig.l. These are

light source

scattering centres light detector

Light Source

This must be monochromatic, that is, its bandwidth must be less than the difference frequency to be measured. Normal gas discharge tubes do not meet this criterion, since they have

a bandwidth of a few hundred to a few thousand MHz. _{(The motion} of the atoms of the gas impart a Doppler shift to the spectral

line)

On the other hand, gas lasers are coherent, and have output at discrete frequencies (determined by the length of the cavity between the mirrors). For instance a Spectra-Physics 124 laser

12-has output at 0.6328 pm or 474 x 10 Hz. This output consists of about 9 discrete frequencies separated by about 214 MHZ)1

This i satisfactory for a liquid velocimeter, where the difference frequency will not'approach this magnitude.

For gas velocity measurement, e.g. of rocket exhausts,6'7 wnere the difference frequency may be greater than. 100 MHz, then a single mode laser must be used (e.g. Spectra Physics 119). This has the disadvantage of a lower power output..

These lasers mentioned above aie helium-neon gas lasers. Argon ion lasers have output in the range 0.4579 pm to 0.5145 pm,

but are more expensive at present fOr a given power output.

Söattering Centres

In tlie published work on the flow of water, 5,8,9,10 the

scatterina centres were. either 0.557 pth polystyrene particles, or they were naturally occuring dust particles (which are present even in ditilled water).

In the work on gas flow, t1e scattering centres were smoke partic1es5, or water mist, alumina, or aluminium ShereS6'7.

The intensity and angular distribution of light scattered in this manner is determined by the Mie theory. _{This requires} extensive computation (see e.g. ref. 12 hich includes a suitable computer program).

Polymer solutions which exhibit drag reduction scatter light, and it is theoretically possible to meaSure the frequency change of this scattered light. It would not then be necessary to add particulate scattering centres.

Appendix I. includes a calculation of the intensity cf light scattered from polymer solutions and by particles and of the

laser power required for a given Signal/noise ratio. It shows.

that particles scatter much more light, and that dilute polymer solutions hardly scatter sufficient light to giveworkable

signal/noise ratjos.

For instance, a I mW laser at O.632.8 urn, when focussed tO a 10pn dia.soot gives sufficient scattered light from one 0.557i.im polystyrene sphere in water to give a 10 dh signal/noise ratio.

C)

gpetctor

Two basic types of detector have been used -- the heterodyne

detector, ard the inter-f erorneter.

A heterodyne detector makes use of the non-linear properties

of a photoniultiplier. The current is proportional to the light intensity (poter) which is proportional to the squar.e of the

electric field strength (at optical freauency). If the scattered light is focussed onto the photomultiplier surface with some

of the unscattered beam, then mixinq will occur at the active

surface. The output current will contain a component at. the difference frequency whióh can be amplified and analysed.

- With this system the thaxirnum frequency of the electrical

signal is about 1000 MHz (set by transit time in the photo-. multiplier), which corresponds to a velocity of some hundreds of

ft/sec. The. resolution or bandwidth can be as narrow as

desired, depending on the spectrum analyser used, and noise in the signal.

A photomultiplier chosen must have hiqh amplification, low noise, and good quantum efficiency. The S-20 coatig is best, with a quantum efficiency of 5% at 0.6328 pm (5% of the incident photons cause the emission of an electron). Suitable tubes are the R.C.A types 7326 and 8645 This method was used

in 3,4,5,8,9,10

The other method of light detection is to use an inter-ferometer in conjunction with a photoinultiplier. This is

equivalent to a tuned filter before the liqh.t detector. Scanning Interferometèrs are available with a 100 mm mirror spacing .which can scan a 1500 MHz hand with a resolution better than 15 MHz

(i.e., finessegreater than 100). With other mirror spacings, the scafl width and resolution change in inVerse proportion. For

instance, for a 50 VHz band, and 0.50 MHz bandwidth, the mirror spacing would be 300 cm or 10 feet, which is impractical.

This method does not reauire that the scattered beam be beaten with some of the unscattered beam, and so alignment is

simpler. It has been used for the measurement of high gas

velocities, see 6,.7 For velocities above 50-100 ft/sec, it is cheaper and simpler to use an intérferometer than optical

hetero-dyning.

d) Optical Components

Fig. 2 shows a more complete diagram of the optical components which would he necessary for the measurement of velocity Of very small elements of a liquid. (similar to

Photomultiplier ( hete r odyne detector cattered

beam

tt Flowing iiqd Pipe

-8-A3 _k2 P2 4 - scattered

beam

Mi

aser

L =lens

A

aperture

M = mirror

P

prism BS = beam splitter Att attenuator

Fig. 2. Doppler Velocimeter for Liquid

Velocity Measurement

In this setup the incident beam from a laser is focussed by L1 to form a spot at the reauired place in the liquid. Light

is scattered from this point and is collimated by L2, and

reflected by M1 onto the photomultiplier surface. The unscattere beam passes through L3 and an attenuator (to adjust its intensity for best heterodyne efficiency), and is reflected by M2 and the

beaim-splitter onto the p1-otomu1tiplier.

M2 is included to make the difference between the path lengths of the scattered and unscattered beams srnall compared

with the laser cavity length. This ensures that the electrical wavefronts of both beams are similarly phased, when a multimode

laser, such as th Spectra-Physics 124 is used.17

Theapertures Al, A2, and A3 are used to reduce the effects of stray liqht both from the laser, and scattered from other

The prisms P1 and P2 are recommended if a velocity traverse of the pipe is required in conjunction with a very small spot size. If the prisms P1 and P2 and the pipe wall have the same refractive index as the liquid, then there will he a minimum of reflection frOm the interfaces (the glass to air interfaces can be coated) o Also if the light enters the prisms at 900 and the scattering angle is 9.00, then a traverse can be carried out without having to refocus or adjust the

system (providing the direction of traverse is parallel to

the. incident beam). Also any shape of pipe section can be

used, since there is no refraction at the. pipe-liquid interface. For air velocity measurements, these prisms are not necessary.

The following optical components have stjll to be chosen Ci) Material of pipe and liquid

Size of apertures and lenses Scattering angle.

Ci) teria] of pipe and liquid

Liquids fall into two classes, those with low refractive index, such as water with n 1.333, and most organics, with a higher refractive index, e.g. n = 1.4 1.5. On the other hand most. glasses have a refractive, index above 1.50. A suitable

match is provided by decalin (decahydronaphtha].efle.)with a

refractive index of 1.476. Toluene would also be. a suitable liquid, hut decalin has advantages of non-inflanimabi:lity, low volatility, relatively low toxicity, and is stable and colorlesS,

It is also a good solvent for polyisobutylene, a polymer which exhibits drag reductiOn.

If the liquid is water, then any glass can be used, but there will be more difficulties due to stray scattered light, and refraction effects. Water was used in 5,8,9,10

-

10.-(ii) Aperture size, and lens focal length..

These variables control the size of the focussed Spot For perfect quality lenses the diameter of the focussed. spot is given by

-d (focal length of lens (Ref. 13)

aperture dia.

where d = dia. of focussed beam

=wavelengthof light

and the aperture diameter iS that of the beam, not of the lens

itself.

If d= 10 pm, A0 = 0.6328 pm, A1 Li mm dia (e.g. Spectra-Physics 124 laser), then the.focal length of the lens must be

14 mm. To allow fOr imperfections in the lens, it should have a slightly shorter focal length, say 10-12 mm.

The focal lengths of and can then be chosen to suit the pipe dimensions, and the apertures chosen so that

(F.L.)3

A2 A3 A1

If a 10 mm focal length lens is not oractical (considering the pipe dimensions, and the required traverse)1 then the laser beam can be expanded in diameter before it entes lens L1.

Appendix II discusses other problems in obtaining a 10 pm

spot. It is essential (at least for liauids) that the light enter the glass at right angles, and that the g1as and liauid have the same refractive index. Even then, the lenses will need to be. corrected for aberrations that occur on entering a flat glass plate.

If the liquid is water, and the refractive Index of the pipe cannot he matched, then à60 im spot is about the smallest that can be obtained (at 300 scattering angle.)

A 50 pm spot should be easily obtained when studying the flow of gases

(iii) Scattering angle,, .e

It is theoretically possible to use any Scattering, angle. If the refractive indices of the liquid and glass are. matched, then an angle of 90° leads to the simplest traversing

system (i.e. refocusing should not be necessary) ,.and small errors due to imprecision in the scattering angle.

If the fluid is water or air then a small scattering angle is better, since it avoids large angles of entry of the light beams into the glass, and gives larger intensities of the scattered beam.

The scattering angle used affects the accuracy of the meaSurementS. This is discussed in a later section.

6. Errors in Doppler Velocimetry

These are of two main kinds, mechanical such as alignment, and information errors, such as broadening of the frequency spectrum.

a. Mechanical .

The main mechanical errors are in the initial setting up of the instrument, so that the three beams have the same focus, in the maintenance of this focus during. a traverse, of the pipe, and in knowing where the focussed spot is, relative to the pipe wall.

12

-If the scattering volume has a linear dimension of 10 urn, then the lenses L1, L2, and L3 should have focal points

coincident to within about 2 pm (i.e

i0000

inches). If there

is any refraction in the pipe (i.e. if the refractive index of the liquid differs from that, of the glass, or if light enters the glass wall obliquely) then this focussing _{will, need to be} reset at every point, of the traverse. _{If the scattering angle} is 90° and 450 prisms are used on the outer pipe wall, and if the pipe and liquid have the same refractive index., then re-focussing should not he necessary.

Another alignment difficulty is that for optimum hetero-dyning both beams hitting the photomu'ltlplier must be the same size, coincident, and parallel. _{AlSo the transmitted and}

scattered beams should he the same length if a'multimode laser is used as the light source.

b. Broadening of electrical signal

Ideally the output of the heterodyne detector at any time will be a signal of one discrete frequency. _{In practice, the} signal will cover a broad range. Of frequencies, _{fdr a number} or reasons, which are discussed helow.

The scattering volume has a finite size. _{or instance in} a liquid system, if TlOO clynes/cm2, and _{10 cm/sec, then} (spot size) x UT = 1 for a .10 pm spot. _{Then in the laminar}

V

sublayer, with y _{u+ for y< 10,. the scattering volume includes} a range of 1 in U, which represents -a signal bandwidth of 10% to 50% of the mean frequenc

Also, light will be scattered from _{points outside. the} nominal scattering volume, or light _{may be scattered more than.} once, causing broadening of the signal.

- .13

Another cause of broadening is the. indefinite knowledge of the scattering angle. To obtain a spot of l0in, it is necessary to use a lens with a semi-included angle of 3.10

(in air), or 2.1° (n n=l.47). This error in the angle occurs in both the. incident beam optiOs and the scattered beam optics, so the scattering angle varies over a range of ± 4.2° from the nominal value. If e= scattering angle, then the output

frequency depends on sin , and. the error is proportional

to cos . For e= 180° the error is negligible, and for 0= 90.,

f/fO4%. For small 0,

f/f0/0, leading to a bandwidth

of 40% when 0= 20°.

Hence, for a l0im spot, small angle scattering cannot be

used. For larger scattering elements, this error is inversely

proportional to the spot size.

Another cause of broadening is diffusion of the. scattering

centres. The half bandwidth at half height is given'4 as

8wDn2

_smn2f,

_{where D is the diffusion constant,} For a solid

° _kT

particle D ₌

_{6p r}

, and for a random coiling macromolecule

D

6 (FgXO 665)

where k = Boltzmann

const.nt,

T =

temperature, = solvent viscosity, r radius, Rg.radiuS of gyration. For polystyrene spheres of 0.557 inn dia. D

0.78 x 108cm2/sec, and the half-bandwidth =. 50 HZ e = 90°)

For a polymer wii-h Rg

500.L D

6.5 x io_8 crn2/sec, and the

half-bandwidth = 450 Hz. This .dfffusion due to. Brownian motion

places a lower limit on the velocity resolution that can be obtained, independent of the siZe of the scattering element.

Signal broadening will be caused by the intermittent nature

of he signal. Consider a 90 volume ppm suspension of 0.55711Tn spheres, and a system with a loinn cubic scattering element. Then the expected number of spheres scattering light = 1.00.

But at any time the number of spheres scattering will he a

random variable with a Poisson distribution, and the electrical signal will appear as an intermittent signal from individual. particles. It is easy to calculate the number of cycles of signal received at the difference frequency from any one particle. if d = size of scattering element, then the time one particle is scattering = d/v1 and the difference frequency

.2 nv

sin 8 . Then the no. of cycles

xc

7

=(2nv

x,. x 14 -30 for 8 900 = 11 for 0 = 300 n = 1.333

= O.E328m

d = l0im

As calculated in Appendix IV, this will cause broadening amounting to half bandwidths at half-I ight.of 3 and l

respectively of the centre frequency.

Signal broadening will also be caused by vibration, and by convection currents in the fluid caused by local heating of the fluid by the laser beam.

Other causes of signal broadening are due to the method Of treating the signal.. A spectrum analyzer has a finite band-width, which may limit the available resolution. If the fluid

flow is turbulent, and the electrical signal is time averaged after being spectrum analyzed, then the electrical signal will he broadened in proportion to the width of th velocity spectrum

15

-c. Signal-Noise Ratio

Even if the system is mechanically perfect, the signal will contain noise from the laser and the photomultiplier, as well as being broadened. Signal-noise ratios have been

calculated by Lastovka15. With 5% quantum efficiency (S-20 photomultiplier), a 1 MHz difference Signal, a 10 KHZ bandwidth

in the spectrum analyzer, a 1 sec. output filter time constant, a heterodyning efficiency of 1 (which. may not : be attained),

and 1. coherence area on the photomultiplier surface, then for asignal-npise. ratio of 10, the Scattered light intensity at the photomuitip1ie must be about

i0'2

watts.

It is shown in Appendix I that this intensity and signal/noise. ratio obtained, using a 500 mW laser, when Scattering light

from. a 10pm spot in a decalin solution of i00:ppui po1ysObutylene of 1,000,000 molecular weight.

A single 0.557 pin dia polystyrene sphere in water will scatter sufficient light when a 1 mW 1ser is focussed to a 10pm spot, at any scattering angle.

5. Description of Instruments Already Built

Ci) Yeh (Columbia)2.

Yei measured velocity profiles in laminar flow in a

22.6 inn tube at Reynolds numbers from 2-6.. He used an

un-focussed beam Of 1.6 inn dia; and scattered light from poly-styrene spheres at an angle of 30°.

(ii) Brown ngineering Co., Inc.

Brown has studied velocimeters for gas3'5 and liquid4'5 velocity measurement. For their liquid velocity measurements,

- 16

at 90°, and the scattering angle 18.5°. They found that it was not necessary to add scattering centres to water as the Huntsville tap water cOntained sufficient dust, etc. The

scattering volume was quoted as 45i.im long (the length is perpendicular to the pipe wall). They have measured both

laminar and turbulent flow at Reynolds numbers from 125 to 12,500, and have measured intermittency at a Reynolds nUmber of 2600.

Brown Engineering Co., Inc. developed this instrument with the aim of marketing jt. This instrument has a geometry

similar to that in4, with a 15° scattering angle (in air), not that shown in5. The latter geometry is essential for

turbulent flow measurements, in order to measure meaningful velocity components, It costs about $5,000 without the laser, or any electronics apart from the photomultiplier and power

supply.

This instrument is now marketed by Applied Lasers, Inc., of Tullahoma, Tenn.

Univ. of Minnesota, Dept. of Mech.

Eng9'1°

Their equipmenthas measured the lamjnar and turbulent flow of water containing polystyrene spheres in

a duct 1 cm. square. They use a very long focal length lens, so the scattering element must be of the order of

1 nun, in size, and use a scattering angle of 30Q in air (with each beam entering or leaving the duct at an angle of 15! to the normal).

Stanford UniV. Dept. of Aero and Astro.6'7

Their instrument was designed to measure particle velocities in rocket exhausts. Consequently it was designed for high velocities (several hundred to several thousand

17

-metres/seö), and uses a scanning interferometer rather than optical heterodyning.

It also uses a large scattering angle (180°) to simplify the optics, and to reduce errors due to measurement of the

scatteting angle. The instrument does not measure a meaningful f low component if the flow is turbulent,

(v) Summary

Most of the instruments built so far have been designed primarily, for laminar flow. Except in6'7 (rocket

exhaust flow) very little attention has been paid to the size of the scattering elerient, which is generally large by comparison with the scale of the turbulent flow near the

wall.

Also many of the

instruments

donot measure a particular velocity component, since either the incident beam or the scattered beam enter the pipe at a right angle to avoid refraction.

7.

Summy

It is possible to measure the velocities of fluids by measuring the Doppler shift in frequency of light scattered

from the fluid.

To use this principle to study the turbulent flow of liquids, it is necessary that the instrument have a spatial resolution of 10 to 50 .im, (at high and low Reynold's numbers

respectively).

To reach the lower limit of 10im, it is necessary to use diffraction limited lenses, and to match the refractive indices

18

-of the glass pipe and the liquid, and to use prisms on the outside of the pipe So that the light enters at right angles. Even then the lenses used must be corrected for aberrations which occur when the light enters the plance glass surface. A resolution of about 60.unis the best that can be obtaired when light enters the pipe at an angle.

A velociineter can be designed to cover a wide-range of velocities, from a fraction of a cin/sec, to gas velocities above 100 ft/sec.

The intensity of the scattered Signal, or the signal/noise ratio of the output depends on the scattering centres use4.

For a signal/noise ratio of 10, a laser power of 500 mW would he reàuired when polymer solutions are used. The design of

this instrument wOuld be difficult. If solid particles are used as scattering centres, then usab1e signals are obtained

from a laser power of 1 InN. (for the case of 0.557 im poly-styrene sphereS suspended in water.)

The. main errors in the measurement are due to the'finite size of the scattering element, afld the finite range of the

scattering angle. The first causes errors becauSe the

scattering element includes regions of different velocities, and the second because the incident and scattered beams are

cones of. light with

finite included angs!

Particle diffusion

Sets a lower limit to the velocity that can be measured.

Other universities and companies have built instruments which give satisfactory results in 1aiiinar flow. They have

generally not been designed for a ma11 scattering element, or to measure velocity components.

19

-8. Conclusions

1. It is very difficult, and probably impractical to design an instrument in which light is scattered from polymer molecules alone. Solid particles scatter much more light, and will, give a cheaper design with much better signal/noise ratios.

2. The instrument is not very satisfactory for the study of the turbulent flow of liquids. The best spatial resolution which can be obtained in aqueous flow, 60 urn, may be too large

at high Réynold's nuiñbers.

3 The instrument seems to he most useful in theStudyof gas flows, where. refraction effects do tiot occur, and where

the turbulent scales are larger. At velocities over 100 ft/sec, the design can be further simplified by. the use of interferometric techniques, rather than optical heterodyning. The instrument

Appendix I

Calculation of Intensity of Scattered Light

(a). Light scattered by Polymer Solutions

The theory of light scattering by polymers is described in polymer text-books, such as

Tanford)6

For light scattered from dilute solutions of small polymer molecules, I 0 NAy A0 20

-(MC

N

Akr

Av

0

when the incident beam is vertically polarized and the incident and scattered beams lie in a horizontal plane.

Collecting the fixed solution and optical terms in

2w2n2(

, we have

per unit Scattering volume,

per unit scatterincy volume.

dn

Substituting n 1.474, = 0.0374, NAV= 6.02 x io23 and

0.6328 urn, then K = 6.2 X 10 9cm2 gin mole/gin2. (e.g. for

polyisobutylene dissolved in decalin).

is

a3

4X0r

The scattering volume. _{_sin} where d

= perture)

for a diffraction limited len9, and

= 10

Then. 32 KM

0

if2 d sine

which indicates that P

A,

-I

-I (sinO)

Mc

Say N w io6, and c = 100 ppm, d lOim, and e = 900

Then P

P0 = 8.0 X 10

A further factor is interference of the light within a polymer molecule, which will reduce the intensity of the scattered beam. This factor is 21 -where

(4nnR

c!4Q

q 2

22

-Typical values of P. (0) are (for n = 1.474,0 = 900, ₌ O.6328i.un)

:Rg (nm) P (900) o

L000

0.02 0.945 0.04 0.806 0.06 0.636 0.08 0.481 0.10 0.358

If the radius of yration of the polymer is 0.075 iirn, then

8.0 x

io2

X 0.516 = 4.1 x

io12

P

S

P0

Assume that the transmission through the optics is 50%,

1'rec -12

then

=2X10

laser

For a signal/noise ratio of 10, a received power of l0 watts is necessary.

Hence the laser power needs to be 500 mW.

The required laser power can be reduced if a higher concentration of polymer is used, or if a shorter wavelength

is used. For instance, a 1000 ppm solution, and a 50 rnW laser

could be used, if the relative viscosity of the solution was not irportant.

This calculation does suggest that scattering from polymer solutions may not be. very practical.

Symbols used:

_is

scattered beam intensity at distance r

= incident beam intensity

n = refractive index. of solvent

M = moleôular weight of polymer

c = concentration Of polymer

(gm/cm3)

= Avogadro's number

A0 = wavelength of light

d = focussed spot width

aperture

= aperture. of lens (unfocussed beam)

PS

power of scattered beam

P0 = power of incident beam

Pg = radius of gyration of polymer moleule

(b) Light Scattered by a polystyrene particle

The. calculation of the intensity of light scattered by

a spherical particle is more difficult than for polymer solutions,

and requires machine computation.

.n example of such a computation

is shown in Fig. 3.

From this graph it can be seen that

scatteriig is mudh more intense in the near forward direction

(at low angles).

For the particular particle size chosen,

scattering is a minimum near 900.

For 0.557

.im polystyrene spheres in water, at 8

90°.

= .0226

p

i'i

\fi \(i.

S

1..SH

S _f 0 P

i JI J%P

0

'0/ \S/\O

2 22

4ir

n

r

71

= .02,26

A2 .0

23

-222

4ir

n

r

per particle

)

/

2'

( ir(aterture)

k 4

)

C C 0) C CD C

0

__

II.

--II

--II

.'ui .'ui

-iuuiut

--'

.iu'i&. .m-:uu

ui.

IIIIi

II

____iII

lE:!tjL1

!

1II!I!HI!IIIIHIUI_IIIIIIIIIII_____

°

I

IUIIih'!i!H!i1IIiHIII

IIIIIIIIIII

iIiiIiiHIHI

1I

rI

IiIi1IP:iI

.

ii

.

..

i.0 iiu

u

iuL

I;IIIII!1

I

I 11111

____u

I UU I

ii ii

ii IiU

..uu.u:uu u

ii III

11111 IIulIIl_

111111 III liii III

IiIIIIIIUIIIII I

i

iuuii:iiuiiu iiuuuui:ui:iiuuunuiIuiIIi 1 I II II III ________I

I_I.IIIuI

6 6 d b 6 b o A ZI( w u o

00

25

-For a diffraction limited lens

4A0 aperture = r wd

-2

1

4

('

(4x

- .0226

₂₂

¶d)

Tj

= .0226 4

244

nird.

Assume 20 transmission through the optics (the refractive index of the glass windows is not matched to that of the fluid)

Then P

=l.7X 10

laser

For p =

io_2

W (to give 10 db S/N ratio)

the laser power must he 0.6 mW.

This shows that even at a large scattering angle of 900, a single 0.557 .im polystyrene sphere in water scatters 1000

times more light than a 100 ppm solution of polyisobutylene in decalin. Part of the reason is the larger difference in refractive index between polystyrene and water than between PIB and decalin. The main reason is that the polystyrene -Sphere is much larger than a polymer molecule.

-For small angles of scattering, say 30°, the. intensity is some 200 times greater, and there should be nO problems in obtaining good signal/noise ratios.

?.ppendix II

Effect on a Focussed Beam of a chnge in Refractive Index of rediur

Suppose a beam of light enters a plane glass wall Of refractive index n at an angle of a. Suppose that the beam is then focussed to point o in the glass, instead of the point I, had the glass not been present.

cot = AIR

Refractive Index

1.00

2G

-GLASS

Refractive Index

n

:ew CO is a line through 0: tan a, o) with a elope of

sin a

A iS the origin of the

coordinate system.

I has coordinates (

0,

-b )

Then sin a

sin

The equation of Co is in2 sin2a -

Sina

/2

.2

yn sin a cos a

/2

.2

in -sin a sin a 27 -- b tan a)

Consider now the point of intersection of this ray, and a neighbouring ray which entered at an angle of (a+)

At 0, the point of intersection

/2

- sin2a ( x-b tan a) sin a

h2-52

_(a+)

cbs (ct+6) - in -sin (a+) sin (ct+6)

/2

.2

in -- sin (a+6)

sin (a+6) x-b tan (6+6)

Sinde 6 is snail, this expEssion can be expanded in _powers

of 6. Then taking the limit as &O, we get

x

n-i

3 =

-(

2Jta

a 2 2 .3/ and y - (n -sin a)

23

n cos a

Hence the coordinates of 0 are

/ ₂

- 2 2

312\

( ( ₂

J

tan3a, -b

fl

Sina)

28

-Consider the following cases. for b .= 5 mn.

Beam enters glass of n = 1.474 at right angles. _{Then for} a 10 pm spot,u varies from -3° to + 3°.

When a = 0° (beam through centre of lens) the coordinates of 0 are (0, - 7.37). That is, the true focus is 7.37 rrn into the glass.

When a 3° (the beam at the edge of the lens) then the coordinates of 0 are (-0.0004, -7.3864).

This means that the light from the edge of the lens is focussed some 16 pm further into the glass (or liquid) than that from the centre of the lens. This could be reduced by suitable design of the lenses.

Beams enter obliquely into a medium of n =1.333 with _a rangin fror' 12° tO 18°. ( e = 300).

The coordinates of the focal point are:

(A thin glass window has been ignored in this calculation.) In this case the focus covers a region about 021 _{n'm, or 27Opm long} in the y direction (i.e.

approx. perpendicular to

_{the pipe wall).}

If light enters the pipw wall at l5 _{(nomina1) it is not}

possible to get a spot less than 60p in an aqueous medium (e.g. with a ranging from 14.5° to 15.5°).

Beam enters a medium of n = 1.333 with _{a =} (450 ±

The coordinates of the focal point are:

a x-coordinate y-coordinate 42 _{-1. 6 -10 . 5} 45 _{-2.2 -11.5} 48 -3.0 =12.7 a Focal Coordinates x _y 12° -0.021 -6.863 15° -0.042 _-6.981 18° -C.075 -7.l32.

If the scattering volume has dimension d, and particles are moving through. it with velocity v, any one particle will be

scattering light for a time = d/v. The difference frequency

=2nv

e

x

sin.

xo

Therefore the no. of cycles of the difference.frequenqy

=

2Zx

sin

)

(

2nd . = sin

0

= 30 for e = 900 = 11 for e = 300 when n = 1.333 0.6328 urn d = 10 urn - 29

In this case the focal area covers more than 2 mm. --it would be bett?r to use the unfocussed parallel laser beam!

This calculation shows that it is not practicable to use a large oblique angle of entry to the fluid. If a large

scattering angle is used, it is essential to use prisms on the pipe wall, and to match refractive indices of the pipe and the

fluid.

Appnc3jx III

Broadening of Signal due to Scatteing from Individual Particles or Molecules

and say f(t) =

A(u)

A(21Tf)

f

0

Then A(u)

) f(t) Sifl Ut dt

iT

N/f

2 ₎

sin 2lrft sifl ut dt

(UN)

-4 f sink-f"

22

2

4irf

u

The centre amplitude is found by taking the limit of ?(u) as

u-42irf

A(2Trf) = urn

A(u)

= urn A(2Trf + £)

.u-*2Trf

c - 0

N

30

-This small number of cycles causes apparent broadening of

the signal.

This is shown by the following Fourier transformation

of the signal.

₍

I

Say f(t) =

.sin 2,rft

for 0<t

N intégral

0

otherwise

A(u) sin utdu

UN

-4f271

r

N(2ir2f2

-

u2)

If u2w f, then

2irN, and the amplitude of sin

fluctuates

through a number of cycles with changes in u.

Hence for a very

approximate look at the spectrum it is possible to set the sin

Then 31 -A(u) ± 4lrf2 A(2nf) - N(u2 - 4w2f2 At half-height, A (u) A(2,rf) and u 2f

Then half-bandwidth at half height

f

1TN

Hence if NI1, the half-bandwidth at half-height is about

3%. of the frequency (e.g. a 10pm spot at e = 30°)... For N=30, it. is Only about 1% (e.g. a 10pm spot at 0 = 90°).

32

-Appendix IV

References Cited

Virk,. p. 5., (1966), "The Tonis Phenomenon - Turbulent Pipe Flow

of Dilute Polymer Solutions " D.Sc. Thesis. Dept of Chem. Eng.,

M.I.T.

Yéh, Y., and Cummings, H. Z., (1964) "Localized Fluid Flow Measurements with an He-Ne Laser Spectrometer." Appl. Phys. Letters 4, (10) 176.

Foreman, J. U., et al(l965),."Measurement of Localized Flow Velocities in Gases with a Laser Doppler .F1oneter." Appl.

Phys. Letters 7

(4)

77.

Foreman, J. W., et al (l966a) "Laser Doppler velociineter for

Measurement of Localized Flow Velocities in Liquids." Proc. IEEE..

::

424..

Foreman, J. W. et al (1966b), "Fluid Flow Measurements with a Laser Doppler Velocimeter." IEEE J. Quantum El. QE-2, (8) 260.

James, R.N. et al (1966), "A Laser-Doppler Technique for the

Measurement of Particle Velocity in Gas-Particle Two Phase Flow." AIAA J. 6(1), 160 (1968).

Morse, H. L., etal (1968), "Development of a LaserDoppler Partiàle Sensor for the Measurement Of Velocities in Rocket Exhausts." AIAA Fluid and Plasma Dynamics Conference, Los Angeles, Calif .,

June 24-26, 1968, Paper No, 68-723.

Berman, N. S., and SantoS, V. A. (1967)., "Velocity Measurements Using the Laser Doppler Technique." AIChE 60th Ann. Meeting New York, N.Y., Nov., 1967, Paper No. 32e.

Goldstein, R. J., and Kreid, D. K., (1967) "MeaSurement of Laminar Flow Development in a Square Duct Using a. Laser-Doppler Flownieter."

Tr. ASME, 3. App. MeOh. 34, (4) 813.

GoldStein, R. 3., and Hagen, W. F., (1967) "Turbulent Flow MeasirementS Utilizing the Doppler Shift of scattered Laser

Radiation." Phys. Fluids. 1Q (6), 1349. ii.. spectra-Physics Model 124 Instructjon Manual.

12. Fahimian, E. Jo, (1967) "Scattering of Radiation by Particle Layers," Ph.D. Thesis, Dept. of Chem. Eng., M.I.T.

33

-James, R. N., et al (1966) "Application of a Laser-Doppler Technique to the Measurement of Particle Velocity in Gas-Particle Two-Phase Flow," Rept. No. AFRPL-TR-66-l19, AD802976. Dubin, S. B. et al (1967) "Observation of the Spectrum of Light

Scattered by Solutions of Biological Macromolecules." Proc.

Nat. Acad. Sd. USA 57, 1164.

Lastovka, J. B., (1967), "Light Mixing Spectroscopy and the Spectrum. of Light Scattered by Thermal Fluctuations in Liquids." Ph.D. Thesis, Dept. of Physics, M.I.T., p. 254

Tanford, C. (1961) "Physical Chemistry of Macromolecules." Wiley, Chapt. 5.

17: Foreman, 7. W., (1967), "Optical Path-Length Difference Effects in Photomixing with Multimode Gas Laser Radiation." Appi. Optics 6 (5), 821.

tBiclassified

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FORM

147"

(PAGE 1) 5NOV65 6) S/N 0101 -80768tI

:;-.

Unclàssified

_Srity Classification

DOCUMENTCONTROLDATA-(Seiurity cta,siftration of title, body of ab,tract .nd indexing annotation

R&D

mu,l be entered when the overall report is classified;

5. OLIGINATING ACTIVSY (CotpoMte ufhor,)

Massachusetts Institute of Technology

Department of Chemical Engineering

is. REPORT 5ECURITY CLASSIFICATION

TTn1ac..jfjed

ib GROUP

Scattered Light.

-3. REPORT TITLE

Fluid Velocity Measurement by Doppler Shift of

4. DEScRIpTIvE NOTES (Type ólreporiiind.lnctu.I?e date.)

Pait of a Proress. ie sort

October .1

1968.

5. AU TISORIS) (First name. middi iniflal. ia.1 name)

William B. Clarke

6. REPORT DATE .

July,

1968

75. TOTAL NO. OF PAqES

33

7b. NO' OF REFI

. 17

- Ba. CONTRACT OR GRANT NO.

ONR Contract No.Nonr-3963(10),

b. PROJECT NO.

_{NR 062-333}

.

d.

s.. ORIGINATOR'S REPORT NUMBERI3)

eb. 0 REPORT NOi$t (Ant. OUI.r numb.ca hatstayb. .utQn.d

fbi. çipofl)

10. DISTRIBUTION STAT EMEPIT

Distribution of this document is unlimited.

0

II. SUPPLEMENTARY NOTES- . $5. iPONSORIUILITARY ACTIVITY

1). AOSTRACT . .

The report -describes the laser Doppler

elocimeter, and the design

necessary to study turbulent flow of liquids where the spatial -resolution rèquired ig

is 10 to 50 *im. -

For the lower. limit., it is necessary to use diffraction

..

limited optics, to match the refractive indices of the pipe and the liquid, and

for the beams to enter and 1ea e the pipe normally.

For a glass pipe

containing water, 60p.m is the best' resolution obtainable.

Much higher "

powered lasers are needed if dissolved poIymer is the scatterer, than if

par.ticle.s are used.

Errors occur due to the velocity gradient in the scattering

element, and to the finite included angles of the incident and scattered beams.

I;

DD

t

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