Calculation of Mass Transfer Coefficients in a

Crystal Growth Chamber through Heat Transfer

MeasurementsJames H. Bell Lawrence A. Hand

Ames Research CenterMoffett Field, California 94035 USA

Abstract-The growth rate of a crystal in a supersaturatedsolution is limited by both reaction kinetics and the localconcentration of solute. If the local mass transfer coefficient istoo low, concentration of solute at the crystal-solution interfacewill drop below saturation, leading to a defect in the growingcrystal. Here, mass transfer coefficients are calculated for arotating crystal growing in a supersaturated solution ofpotassium di-hydrogen phosphate (KDP) in water. Since masstransfer is difficult to measure directly, the instantaneousdistribution of the heat transfer coefficient on a heated scalemodel crystal in water is measured using temperature-sensitivepaint (TSP). To the authors' knowledge this is the first use ofTSP to measure temperature distributions in water. Thecorresponding mass transfer coefficient is then calculated usingthe Chilton-Colburn analogy.

Measurements were made for three crystal sizes at tworunning conditions each. Running conditions include periodicreversals of rotation direction. Heat transfer coefficients werefound to vary significantly both across the crystal faces and overthe course of a rotation cycle, but not from one face to another.Mean heat transfer coefficients increased with both crystal sizeand rotation rate. Additional experiments show that continuousrotation of the crystal results in about a 40% lower heat transfercompared to rotation with periodic reversals. The continuousrotation case also shows a cyclic variation in heat transfercoefficient of about 15%, with a period of about 72 times therotation period. Calculated mass transfer coefficients werebroadly in line with expectations from the full-scale crystalgrowth experiments.

I. NOMENCLATURE

c ConcentrationCP Specific heat at constant pressureD Mass diffusivityEc Eckert numberg Gravitational accelerationGr Grashof numberh Heat transfer coefficientkc Mass transfer coefficientJ Chilton-Colburn numberL Length scale (rot. axis to corner of crystal)mii" Mass fluxNu Nusselt numberq"f Heat flux (power/area)P Period (also pressure)

PrReScShSttTu

Ux

x

FMWx

KDPNIFTCRTSP

a

pv

0)

Q

Prandtl numberReynolds numberSchmidt numberSherwood numberStrouhal numberTimeTemperatureVelocity (of fluid)Peripheral speed (of corner or crystal)Position vectorWidth of crystal

SubscriptsFull-scale conditionModel-scale conditionWall (solid-surface) conditionBulk fluid condition

AcronymsPotassium di-hydrogen phosphateNational Ignition FacilityTemperature Coefficient of ResistanceTemperature-sensitive paint

Greek SymbolsThermal diffusivityDensityKinematic viscosityViscous dissipation functionCrystal or model rotation rateMaximum rotation rate, also ohms

II. INTRODUCTION

A. The NIF Crystal Growth FacilityThe National Ignition Facility (NIF) at the Lawrence

Livermore National Laboratory (LLNL) uses a set of largelasers to heat and compress test samples to conditionsappropriate for the ignition of nuclear fusion reactions. TheNIF requires large crystals of potassium dihydrogen

11-4244-1600-0/07/$25.00 ©2007 IEEE.

phosphate (KDP) for frequency doublers and Pockels cells. Atypical crystal is shown in Figure 1.KDP crystals of this size are grown in specialized facility

at LLNL. The crystals are grown in large (41" (104 cm) dia x59" (150 cm) high) cylindrical tanks containing a solution ofKDP in water. Seed crystals are mounted on a platform whichspins in the solution bath as the crystal is grown. Theplatform does not spin at a constant rate. Instead, it follows arotation cycle diagrammed in Figure 2. Starting from rest, theplatform accelerates to a specified rotation rate, remains atthat rate for a fixed time, and then decelerates to rest. Afterbriefly pausing at zero rotation rate, the platform then followsthe same spin profile in the opposite direction. The "cruise"rotation rate is 75 rpm initially, and is decreased in steps to15 rpm as the crystal reaches its maximum size. This rotationprofile has been developed through trial and error. In somecases the growing crystal develops imperfections and must bediscarded. Further optimization of the rotation profile, both toimprove growth rate and reduce defects, is desirable.

point on the surface, the solution concentration may dropbelow saturation locally, causing growth to stop at that pointand producing a crystal defect. The present experimentattempts to determine the instantaneous distribution of masstransfer coefficient across the crystal face in order todetermine if it is sufficient to allow crystal growth.Mass transfer coefficients are difficult to measure

experimentally. However heat transfer coefficient can bemeasured much more readily, and under many conditionsheat transfer coefficients can be used to estimate masstransfer coefficients. The theoretical basis for this assertion isdeveloped in the following section. The method for makingheat transfer measurements is diagrammed in Figure 3. Aplastic-block scale model of a crystal is placed on a rotatingplatform in a scale model of the crystal growth chamber.Resistance heaters on the faces of the block generate aconstant heat flux into the surrounding fluid. The surfacetemperature of the faces (Tw) is measured with temperature-sensitive paint, and the heat transfer coefficient, h is given bythe relation h(x, y, t) = I/(T. (x, y, t)-TX ) . Details ofthe measurements are elaborated on in the sections onequipment and experimental procedure.

Co(t) c

PC Board Heater( q" = 0.4 - 1.5

W/cm2)

* To

TW (x,y, t)(Measuredwith TSP)

Figure 1. A high quality 50 cm KDP crystal produced for theNIF.

Figure 2. Notional rotation rate vs time schedule for crystalgrowth experiment. 0 is the maximum, or "cruise" rotation

rate The acceleration period, cruise period, and stop period areall specified as fixed fractions of the total period P.

The growth of crystals in a supersaturated solution dependson two independent factors. Reaction kinetics controls therate at which atoms are added to the crystal. Growth mustoccur at the same rate across the entire crystal face. At thesame time, the concentration of solute must be above thesaturation level at all points on the surface. At each point onthe crystal surface, growth extracts material from the solutionwhich must be replaced by convecting/diffusing new solute tothe surface. If there is insufficient convection/diffusion at a

Figure 3. Diagram of heat transfer experiment. Surfacetemperatures are measured with temperature-sensitive paint onelectrically heated model that rotates at rate Co(t) in fluid with

bulk temperature T,

B. Calculation of Mass Transfer from Heat TransferMeasurements

Consider two experiments. In the full-scale experiment (F) amass transfer measurement is desired while in the model-scale experiment (M) a heat transfer measurement can bemade. It is assumed that the heat transfer experiment isconducted in a single-species fluid, so species transfer isirrelevant, while the mass transfer experiment occurs atconstant temperature, so heat transfer is irrelevant.Mass transfer is governed by the non-dimensionalized

equations (1, 2 and 3) below, while heat transfer is governedby Equations 4, 5, and 6. These equations are forincompressible flow in an inertial (non-rotating) coordinatesystem. No temporal or spatial averaging has been applied.It is assumed that concentration and temperature variationsare small enough that the fluid density varies approximatelylinearly with them. Other fluid properties are assumedconstant. Free-surface effects are excluded.

2

Mass Transfer (Momentum & Species)

au *St + (u .V*)u*

Et*Dc*

St +u Vc =At*

1 V*2U* _V*P*Re

1 *2C*Re Sc

B.C. at walls (x* = x ): u

Gr *g~c g

Re2

XW, (n*V*c*)W1

0

Heat Transfer (Momentum & Energy)

au *St +(u .V*)u*

At*

T* *St

a+u.V

1 V*2U* _V*P*Re

Gr *g*Re2 T*g

1 V*2T*+EcVRe Pr Re

B.C. at walls (x* = x): u=, (n v*T*)W1

0

(1)

(2)

(3)

(4)

(5)

(6)

Here, u, P* T, c, and t* are non-dimensionalized velocity,pressure, temperature, species concentration, and time,respectively. g* and ID are the non-dimensionalizedgravitational acceleration and viscous dissipation function,respectively. St, Re, Gr, Pr, Sc, and Ec are the Strouhal,Reynolds, Grashof, Prandtl, Schmidt, and Eckert numbers,respectively. x is a non-dimensionalized position vector andn is a unit normal vector. It should be noted that the non-dimensionalized pressure incorporates a gravitationalhydrostatic term:

* p+p= gz (7)PJU2

where Pci. denotes the density of the undisturbed fluid and z ispositive upward.The boundary conditions (Equations 3 and 6) merit some

explanation. The solid surfaces (denoted by convention as"walls") of the model and full-scale systems must be in thesame scale positions at the same scale times:

XWM(tM) XWF(tF) x (t) = xW (t)/ L t = t IP (8)

Here, L is a reference length, defined as the radial distancefrom the axis of rotation to a corner of the crystal; and P isthe period of the rotation cycle (Figure 2). Also, the constantheat flux at the surface of the model crystal corresponds tothe constant mass flux at the surface of the full-scale crystal.The non-dimensional concentration and temperaturegradients at (and normal to) a solid surface have the value ofunity if there is mass flux to or heat flux from that surface,and the value of zero in the absence of such flux. Thus, the

unity value applies to the active wall (all of the crystalsurfaces that are in contact with the fluid), and the zero valueapplies to the passive walls (all other parts of the apparatus-which are assumed both impenetrable and adiabatic that arein contact with the fluid).

Inspection of Equations 1-6 reveals that the heat transferequations are nearly identical in structure to the mass transferequations, with the non-dimensional temperature T* takingthe place of the non-dimensional concentration c . The onlystructural difference is the viscous dissipation term in theenergy equation (5). For accurate physical modeling, thisnonconforming term must be negligible (see below).The structurally equivalent sets of equations can be

rendered functionally equivalent by matching parametervalues. The full-scale mass-transfer experiment (F) ischaracterized by the non-dimensional parameters StF, ReF,GrF, and SCF, while the model-scale heat-transfer experiment(M) is characterized by the non-dimensional parameters StM,ReM, GrM, PrM, and ECM. The functional forms of Equations1 and 4 would be identical if StM=StF, ReM=ReF, GrM=GrF.Furthermore, the functional forms of Equations 2 and 5would be identical if, in addition, PrM = SCF, andEcM<<]PrM. If all of these conditions were satisfied, andthe boundary conditions of the two systems were equivalent,then the entire non-dimensional temperature field in themodel experiment would be identical to the entire non-dimensional concentration field in the full-scale experiment:

Kk(T -LT

41qL )T* ~* (D(c )) (9)

where, k is the thermal conductivity of the fluid, D is themass diffusivity of solute in the fluid, 4"andh" are thefluxes of heat and mass at the crystal surface, respectively.However, of primary interest here, are the conditions on theactive wall (i.e. the crystal surface):

k(Tw TJ)4"'L WM C* =D(cW -CW)FWF fill L ) (10)

Both sides of Equation 10 can be inverted to show that it isequivalent to the following expression that equates theNusselt number (Nu) to the Sherwood number (Sh). [Theseare non-dimensional forms of the heat transfer coefficient (h)and the mass transfer coefficient (kc), respectively.]

NUM &(hL)Nk M

I I -rkcL

TWM CWF KDFShF ( 11)

Assuming that all of the specified conditions were met,Equation 11 could be used to obtain the mass transfercoefficient in the full-scale experiment (kcF) from ameasurement of the heat transfer coefficient in the model-

3

scale experiment (hM). However, it is worthwhile to discussthese conditions in detail, since they have importantimplications, and since several of them can be relaxed inpractice.Matched Strouhal numbers:

1

QMPMLM

UMPM-Stm =StF LF

FUFPF1

QFPF

Here, Q is the nominal ("cruise") angular speed (Figure 2)and U is the equivalent peripheral speed. This requirementestablishes consistency between the speed and time scales.Notice that it is equivalent to matching the number ofrevolutions in the rotation cycle. This is important forreplicating large-scale flow features (e.g. swirl, vortexformation) and is thus "non-negotiable."Matched Reynolds numbers:

Q L2M F- RemVM

ReF QFLFVF

Here, v is the kinematic viscosity of the fluid. Thisrequirement matches the ratio of inertial and viscous forcesand is thus the sine qua non of fluid mechanics experiments.It ensures that the momentum boundary layers in the model-scale experiment have the same scale thickness as those in thefull-scale experiment. However, in the complicated physicalsituation considered here, matched Reynolds numbers may becritical for accurate replication of other flow phenomena (e.g.laminar-to-turbulent transition, turbulence scale & intensity,boundary-layer separation & reattachment) as well.Combining Equations 12 and 13:

PMPF

L( )2 )-

imposed heat flux, ,Bc is the fluid coefficient of solutalexpansion, and Ac is the concentration change (across theboundary layer) resulting from the imposed mass flux. Theratio GrlRe2 can be easily recognized as a ratio of buoyancyforce per unit mass to centrifugal force per unit mass.Matching the full-scale value of GrlRe2 in the model-scaleexperiment replicates the relative contributions of free andforced convection. This is possible in the present context,because the elevated temperature at the surface of the modelcrystal reduces the fluid density, as does the depressed soluteconcentration at the surface of the full-scale crystal.Moreover, the full-scale GrlRe2 values can be matched usingmodel parameter values that are within the realm ofexperimental practicality. However, approximate analyses ofboth the model and full-scale experiments indicate thatbuoyancy effects are relatively unimportant. In addition,CFD simulations of the crystal experiment with buoyancyterms turned on and off show no detectable difference [1].The buoyancy terms in Equations 1 and 4 are thus presumednegligible and the requirement to match them is therebyeliminated.

Negligible viscous dissipation. In the present situation(assuming PrM near unity), it can be shown that the ratio ofthe first and second terms on the right-hand side of Equation5 is approximately equal to the ratio of heat transfer throughthe boundary layer to heat generation by viscous shear withinthe boundary layer. Consequently, if the motor powerrequired to rotate the crystal alone is small relative to theheater power, then the first term will certainly dominate andthus the second (viscous dissipation) term can be neglected.Using the rotation torque data provided by Robey & Maynes[2] to estimate the average mechanical power per unit surfacearea of the model crystal, the requirement for neglectingviscous dissipation can thus be quantified:

(14) »M >> WM - 0.07PMUm = 0.07PM(QMLM) (16)

Note that the required rotation rate in the model-scaleexperiment varies as the inverse square of the model scalefactor, while the period of the rotation varies as the square ofthe same parameter. Therefore, the angular acceleration rateof the model crystal varies as the inverse fourth power of thescale factor. This turns out to be an important constraint onthe model experiment apparatus.Matched Grashof numbers (with matched Reynolds

numbers):

QgfTAT _ Gr KGr]Q2L ) Re2)M Re2)F

g/8cAcJ- Q2L

F

(Example: For UM=2m/s in water, WM 0.06 W/Cm2)Notice that the mechanical power per unit area (and thus, therequired heat flux) increases with the cube of the peripheralspeed. Consequently, for matched Reynolds numbers, therequired heat flux varies with the inverse cube of the modelscale factor. Excessive heat flux must be avoided because theresulting large temperature differences violate the constant-fluid-properties assumption employed above.Matched Prandtl and Schmidt numbers:

VM -Pram(15)

Here, g is the gravitational acceleration, PT iS the fluidcoefficient of thermal expansion, AT is the temperaturechange (across the boundary layer) resulting from the

SCF VF

DF(17)

Here, a is the thermal diffusivity of the fluid, D is the massdiffusivity of solute in the fluid, and the kinemaic viscosity,v, can be interpreted as the momentum diffusivity of the fluid.Equating the model Prandtl number to the full-scale Schmidt

4

-2

QM Lm vmQF LF VF

number scales the model thermal boundary layer to match thefull-scale diffusional boundary layer (assuming matchedReynolds numbers: ReM=Re). This is clearly desirable, butvery difficult to achieve in the present situation.The Schmidt numbers of the KDP solutions in the full-

scale experiment are large (SCFT700 for the 15-cm and 30-cmcrystals and SCF-2500 for the 60-cm crystal). By contrast,water has a Prandtl number -7 at room temperature. Fluidswith Prandtl numbers in the 700-2500 range are extremelyviscous. If they were used, astronomical model speed andpower would be required to achieve a matched Reynoldsnumber. Therefore, in the present situation, matching of thePrandtl and Schmidt numbers is not experimentally practical.An alternative approach employed in the present study is to

use the Chilton-Colburn analogy to correct for themismatched Prandtl and Schmidt numbers. The Chilton-Colburn analogy is an empirical correlation that is applicableto a wide variety of flows. It is usually written in thefollowing ("J-Factor") form:

NuM =. ~ShF (18)RePr/ JM JF Re SC'13

[Note that this equation reduces to Equation 11 if ReM= ReFand PrM= SCF.] The idea here is that each flow has its own Jvalue that remains approximately constant over a rangeReynolds, Prandtl, and Schmidt numbers. In additional to theprovision for unequal Prandtl and Schmidt numbers, there isalso the option of correcting for mismatched Reynoldsnumbers. Although this might be reliable in some simplephysical situations (e.g. fully developed turbulent tube flow),it seems highly untrustworthy in the present context.However, in the case of data with slightly mismatchedReynolds numbers, it is probably better to apply thecorrection instead of simply ignoring the discrepancy.Equation 18 can be rearranged to yield a heat-transfer-to-mass-transfer conversion factor (with the handy-but-dangerous Reynolds number correction included):

kCF DF LM ReF FSC (19)hM kM LF ReM YPrM)

The following alternative form is mathematically equivalent:

kC.F 1 vFLM ReFi0 PrF (20)hM (pCP)M VM LF ReM SCF)

The Chilton-Colburn analogy has been experimentallyverified over a range of 0.6<Pr<60 and 0.6<Sc<3000, whichencompasses the Prandtl and Schmidt number ranges of thepresent experiment. However, the exponents (1/3 in Equation19 and 2/3 in Equation 20) are somewhat arbitrary. Slightlydifferent values have been found to fit the data better in

specific situations. Hence, there is some uncertaintyregarding the appropriate exponent value for the presentapplication. Consequently, the accuracy of the computedmass transfer coefficient will benefit if the ratio SCF PrM isminimized to the extent practicable.The Chilton-Colburn analogy is not commonly applied

rotating or unsteady flows. It has, however, beensuccessfully used to correlate mass-transfer between rotatingcylinders. Also, the time scale of the unsteadiness in thepresent application is long by comparison to those ofboundary layers and turbulence. The heat and mass transferprocesses are thus arguably quasi-steady.

C. Temperature-Sensitive Paints

In the present experiment, heat transfer coefficients werededuced from instantaneous temperature measurements onthe surface of a model crystal subject to constant uniformheating from the interior. Surface temperature measurementswere made with temperature-sensitive paint, some propertiesof which are briefly described below.

There exist a broad range of luminescent materials whoselight emission (in response to excitation by light of theappropriate wavelength) is sensitive to temperature. Furtherdetails of this phenomenon are well-covered in several reviewpapers [3,4]. To first order, the brightness of a TSP isexponentially dependent on temperature, i.e., as

I1I2

eA(TI -T2) (21)

where I1,I2 are the light intensities emitted at temperaturesT1, T2, respectively, and A is a sensitivity coefficient.Typically luminescence brightness decreases withtemperature, so A is generally negative. Equation 21 can alsobe written as

I1I2

(1+ k)(T2-T) (22)

where k is understood as a (generally positive) sensitivitycoefficient. In this formulation, the material's temperaturesensitivity is naturally expressed in terms of a fractionalchange in brightness per degree of temperature change. Tomeasure temperature, the luminescent material is mixed intoa coating which can be applied to the surface of interest. Thepainted surface is then illuminated with excitation light andtwo images are taken - one with the surface at a knownreference temperature and the other with the surface at theunknown test condition.

III. EQUIPMENT

The model scale crystal experiment must match theoperating conditions of the full scale crystal growth chamberfor the six cases of interest summarized in Table 1. Therequirement to match Reynolds and Strouhal numbersimposes important constraints on the design of the modelscale experiment. Water is the natural choice for the working

5

fluid in the model scale experiment, and the high rotationalvelocities and accelerations which occur at model scalesmuch below 12 present several design problems. The higherrotation rates would require sub-millisecond flash durationsto properly freeze the model in the TSP images. Also, morefrictional heat would have to be disapated in a smallervolume of water, adding another significant heat sourcebesides that from the surface heaters. Thus it was decided toconstruct the model scale crystal growth tank at 12 the fullscale size, using water as the working fluid.

Considerable thought was given to the intriguingpossibility of a heavy gas as the working fluid. The primarycandidate was octafluorocyclobutane (C4 F8, MW=200). Thisgas is non-toxic, non-flammable, non-corrosive, andcommercially available (as Refrigerant C318) at low cost. Itskinematic viscosity (v _10-6 m2/s) is comparable to that ofwater, but its density (p -9 kg/m3) is two orders of magnitudelower. Thus, matched Reynolds numbers could be achievedwith much less power and perhaps smaller models as well(see Equation 16). However, the Prandtl number (Pr -0.8, as

is typical for gases) is only about 1/8 that of water.Therefore, the ratio SCF /PrM would be about eight timeslarger and the accuracy of the computed mass transfercoefficient could thus be compromised, as discussed above.Another potential problem relates to the thermal

conductivity of the gas (k 0.01 W/mK), which is only about2% that of water. Assuming fixed values of JM and theReynolds number, the Nusselt number for the gas would beabout half that for water (see Equation 18). Then, for a givenmodel size, the heat transfer coefficient and thus themaximum heat flux-would be about a hundred timessmaller (see Equation 11). Transient conduction of heat intothe interior of the crystal model would then be important.The models would probably need cores made of plastic foam(instead of solid plastic) to be sufficiently adiabatic.For these reasons, the heavy-gas idea was abandoned

with regret. Even the proposed gas level sensor (a verticalstripe of oxygen-quenched luminescent paint [3,4] on theinside of the tank wall) would have been an interesting andnovel experiment.

TABLE 1.OPERATING CONDITIONS FOR FULL SCALE AND MODEL SCALE EXPERIMENTS. MODEL SCALE EXPERIMENT IS DESIGNED TO MATCH FULL-SCALE REYNOLDS AND

STROUHAL NUMBERS. MODEL AND FULL SCALE CASES MATCH AS FOLLOWS: 1M<->1F, 2M<->2F, 3M<->3F, 4M<-4F, 6M<->1F, 7M<->2F, 8M<->5F, 9M<->6F. CASE 5M ISCONTINUOUS ROTATION AT CONDITIONS CORRESPONDING TO CASE 4F, AND CASE 1OM IS CONTINUOUS ROTATION AT CONDITIONS CORRESPONDING TO CASE 5F.

Full- Crystal Kinematic Acceleration Max Rotation Reynolds StrouhalScale Size (cm) Viscosity (rpm/min) Velocity Cycle Number Number

( 2/,, {m DrrAI\ n 1 C 1

60.0 1.60.0 1.

xl 0'

xl106200

2001535

5656

202 88

471 205

61V1 IA 0.56x1U| 2.4u 113 3U. I ) zI1 1 69M 7.4 0.56x 10-6 2550 169 30.1 87 5331OM 7.4 0.56x 10-6 0 113 58

A. Crystal Growth Chamber Model

A roughly 12 scale model (actually 49.40 o) of the LLNLcrystal growth facility was constructed, as illustrated inFigure 4. Figure 5 shows photographs of the model indicatingits major components. The design philosophy was toreproduce the LLNL growth chamber's physical parametersas closely as practical, while maximizing optical access andallowing model heating. The model crystal growth chamber

consists of four parts: the tank itself, the tank supportassembly, the model platform, and the motor mount. Theoptical system, consisting of the TSP camera and flashlamps,is fixed to the same base as the model crystal growthchamber. The components of the model growth chamber are

described separately below.One significant difference is that the LLNL facility has a

free fluid surface, while that of the model chamber can berestrained by an acrylic top. Despite its relatively large scale,

6

F 3F4F

the higher rotation rate of the model crystal does lead to asignificantly larger amount of surface displacement in themodel system compared to the LLNL facility. It was not clearinitially whether restraining the surface displacement with asolid boundary would be a less significant change thanallowing a free surface with greater displacement. Once themodel was constructed, visualization of the flow indicatedthat the closest match to the full scale facility was obtainedby keeping the acrylic top right at the water surface.

wiring Motor and support with Plexiglas andHeatero slip reduction gear aluminum rotationthroughslip plat-form

ring < s /

/Flashlamp (1 of 2)

I CCD

3/8" (1.0 cm) at the edges. The base is cemented to thecylinder section and the interior is polished to produce asmooth joint between the sections. A ring is cemented to thetop of the cylindrical section of the tank to act as a lip. Figure6 shows a cross-section of the tank giving its dimensions.Tank Support Assembly: The tank support consists of a

3' x 3' (91.4 cm x 91.4 cm) square piece of plywood with a20" (50.8 cm) hole in the center, which supports the tank byits lip. The plywood collar is itself supported by fouraluminum columns which are mounted to an optics table. Theoptics table also serves as a base for mounting the camera andflashlamps. In addition to the plywood collar, the tank is alsosupported at the base of the hemispherical section. Thissupport is necessary because the cemented joint between thehemispherical and cylindrical sections cannot support theweight of water when the tank is filled.

\NWood and aluminumsupport structure

Acrylic tank (49.4% scale)

Figure 4. Side view diagram of model crystal growth chambershowing support structure, tank, optical system, crystal platform,

drive motor, and crystal model.

Figure 5(a). Rear view of model crystal growth chamber showingtank, support structure, and rotation stage with small crystal

model. Rotation stage fins are painted black. Yellow tape holdsintensity calibration coupons.

Tank. The tank is made from clear acrylic and consists of acylindrical section of 20" (50.8 cm) outside diameter, 20"(50.8 cm) height, and 3/8" (1.0 cm) thick, attached to amatching hemispherical base. The hemispherical base isblown from l/2" (1.3 cm) thick acrylic. The blowing processreduces the thickness of the hemispherical section to roughly

Figure 5(b). Front view of model crystal growth chambershowing flashlamps and camera in foreground.

Figure 5(c). Side view of model crystal growth chamber lookingdown. Image shows small crystal model on rotation stage.

Bottom tank support is visible beneath.

7

4-22" (55.9 cm) -|*l

Fr-vcm)

/8" (1.0 cm) 1/2"

12"(30.5 cr

7/8" (2.2 cm)

Figure 6. Cross section of model crystal growth tank.

Model Platform: This assembly consists of two 17.5"(44.5 cm) diameter acrylic disks joined by three fins, asshown in Figure 6. The lower disk is thicker than the upperdisk to match the full scale platform. The aluminum fins aremounted at 120° intervals. The fins have an elliptical crosssection to minimize drag irrespective of rotation direction.The top disk is attached to a hollow aluminum shaft which isturned by the motor. The bottom disk has a pattern of boltholes which allow crystal models of different sizes to bemounted to the platform. In theory the model could bemounted to the platform at any angle. In the full scale crystalgrowth chamber, however, seed crystals are always four-sided, and always mounted so that one side directly faces afin. Model crystals followed this practice.The platform contains wiring for power and/or signal

transfer. Each fin has a channel cut in it for wiring, and eachplate has three channels leading from the center to the fins atthe edges. Wires can be routed from the crystal mount at thecenter of the lower disk through the fins and along the upperdisk to the hollow drive shaft. The wiring channels weresized to allow the installation of up to 12 strands of 10 gaugewire. Currently 6 strands of 12 gauge wire are installed. Thegrooves are filled with fiberglass putty to hold the wires inplace and present a smooth surface to the flow. A Mercotacslip ring mounted on top of the hollow shaft provideselectrical continuity despite the platform's rotation.Motor. The platform is spun by a Parker Compumotor TS

42B stepper motor, rated at up to 400W and 16 Nm oftorque. The motor turned the platform through a 3:1 reductiongear using a toothed belt. The motor drive has a digitalinterface and can accept arbitrary motion control instructions.An optical encoder mounted on the motor drive shaft wasused to monitor the position of the rotation platform. Theprecise speeds and timings of the platform rotation weredetermined by a motion control program running on apersonal computer. This program also controlled theoperation of the flashlamps and camera.

3"(7.6 cm) 17.5"

4-+ (44.5 cm)

Figure 7. Side and top views of rotation platform.

B. Crystal ModelsThree crystal models were made; each following the same

construction procedure. The actual crystals grown at LLNLvary considerably in shape [5,6]. However, only a single,representative shape was chosen for the present experiment.This shape is parametrically defined as shown in Figure 8.Thus the three models are geometrically similar and differonly in scale. The models are sized to represent 15-cm, 30-cm, and 60-cm full-scale crystals.

0.77X43

.< ~~~~~~~~~~~p-x

Figure 8. Side view of crystal model showing shape parameters.For a given base width, X, the total model height is 77% ofXPyramid facets are set at an angle of 43° to the horizontal.

Each model is built from a PMMA (Plexiglas) core, witheach face of the core covered by a thin heater board. Thecores are made slightly undersized to accommodate theadditional thickness of the heater boards. The cores havehollow centers which serve as wiring compartments.The heater boards are made from thin FR4 glass-epoxy

circuit board material clad with copper on one side, on whichtraces are etched in a serpentine pattern. Figure 9 shows atypical etching pattern, in this case for a pyramid face on the30-cm scale crystal model. A trace pitch of 0.1" (2.5 mm)was maintained for all heater boards, so the number of tracesvaries with the size of the board. The rectangular prismboards are similar to the pyramid board shown in Figure 9.Two sets of heater boards were manufactured. The first set

8

All traces are terminatedwith a 0.03" diameter

el etched disk <

Etched line between

I

00

11

_

Cornerangle =53.820

Drill 0.05" diameterhole at lower corners of

triangle for wiring.I' 0.08"

Figure 9. Heater board etching pattern for triangular face of30-cm scale crystal model. Trace width is fixed. Number oftraces varies with board size. Holes in lower corners allow

wiring through back of heater board.

included heater boards to cover the 30-cm and 60-cm scalemodels. These boards were made using 0.008" (0.2 mm)thick FR4 circuit board with 0.00017" (4.3 ptm) thick coppercladding. These boards displayed a tendency to burn outwhile under load. A plot of the theoretical current andequipotential lines showed excessive current densities on theinsides of the 180° turns in the serpentine pattern at the edgesof the board. Later, a second set of heater boards was made tocover the 15-cm scale model, as well as the 30-cm scalemodel for additional experiments. For the second set ofboards, etched disks were added at the ends of the linesbetween the traces, and the copper thickness was doubled to0.00035" (8.6 ptm).* No bumouts occurred with this set ofboards.

Figure 10. 30-cm crystal scale model with heater boardsattached. Serpentine pattern can be seen on faces. Power leadsare attached to inward-facing sides of boards at corners. Leadspass through holes in PMMA core to interior wiringcompartment. Protruding ends of power leads are visible atcorners of model. These are trimmed before application of TSP.

Actual copper thicknesses were specified in industry-standard units of ounces of copper per square foot of area.Thicknesses of 0.00017" and 0.00035" correspond to 1/8 and1/4 oz Cu / sq ft, respectively.

Figure 11. 30-cm crystal scale model after application ofTSP. Black dots are photogrammetry reference marks.

Power leads were wired to the back sides of the heaterboards through small holes drilled in the boards at theendpoints of the serpentine traces. The boards were thenglued to the faces of the PMMA cores using contact cement.Figure 10 shows the 30-cm scale model crystal with theheater board faces attached. Once the boards have beenattached to the core, the model can then be painted with TSPas shown in Figure 11.

(a) 60-cm scale model.Total R = 8.4 Q.

+(b) 30-cm scale model(thin copper heaterboard). Total R= 8.4 Q.

(c) 30-cm scale model,, + (thick copper heater

board). Total R =17.2 Q.(d) 15-cm scale model.Total R= 5 Q.

Figure 12. Wiring configurations used for crystal models, and totalresistance of heater boards. Configuration (a) is parallel. Config-uration (b) is series/parallel. Configurations (c) and (d) are series.

The heater boards were driven by an available 2.6-kWvariable autotransformer capable of providing 0-130 VACand up to 20 A. Within the model, it is possible to wire theindividual heater boards in series, parallel, or anycombination thereof. The wiring patterns and total resistancevalues for the three models (including both sets of heaterboards for the 30-cm scale model) are shown in Figure 12.Generally speaking, one would like to have the widestpossible range of available heating power, which in turnimplies an optimum total resistance of 130/20 = 6.5 Q. Thefirst set of heater boards for the 30-cm and 60-cm scalemodels were wired to most closely approach the optimumtotal resistance. The second set of heater boards for the 15-cmand 30-cm scale models was wired in series to simplifyinstallation. For the 15-cm scale model, series wiring gives

9

nearly optimum resistance. For the 30-cm scale model,sufficient power was available even with a non-optimalresistance.At 60 Hz, the measured phase shift between the heater

voltage and current was less than 2° (in both air and water),indicating an almost purely resistive impedance and a powerfactor of essentially unity.

C. Camera andLamp SystemCamera. A scientific-grade camera based on a

thermoelectrically cooled SITe 1024 CCD (charge-coupleddevice) and a Princeton Instruments ST-138 controller wasused in this experiment. The 1024X1024 pixel, back-illuminated CCD has a high quantum efficiency of nearly80% and a high full-well capacity of roughly 300,000photoelectrons. The controller's 16-bit analog-to-digitalconverter reads out the CCD at 430 kilo-samples/sec. Thecamera lens was equipped with a bandpass interference filter(passband = 620 nm, full-width half-maximum = 10 nm)chosen so that the passband of the filter matched the emissionpeak of the TSP. The camera has a frame rate of 1/3 seconds.A 50mm fl.4 lens was used for cases IM - 5M. An 85 mmfl.8 lens with a macro ring was used for cases 6M - 7M, anda 135 mm f2.8 lens, also with a macro ring, was used forcases 8M- IOM. Lens apertures varied from f4 to f8 duringthe experiments. The camera is a frame transfer camera, withlight access to the CCD being controlled by a mechanicalshutter with an opening time of about 15 milliseconds - tooslow to freeze the motion of the platform.The flashlamp consisted of a single Norman 4000 Joule

flash power supply driving two lamp heads. Each lamp headwas driven at 1000 Joules. Flash duration at this power ratingwas specified by the manufacturer to be 4 milliseconds. Eachflashlamp was equipped with a tempered Schott glass BG-25filter to absorb light emitted by the lamp at the paint emissionwavelength, while transmitting light at the paint excitationwavelength. The flashlamp power supply requires 7 secondsto recycle.Image acquisition timing was determined by a motion

control program running on a personal computer. Asoriginally set up, the computer simply issued a command totake an image at the desired platform position. Since thecamera shutter was much slower than the flashlamps, it waspre-opened 0.05 seconds prior to the platform reaching thedesired position, and the flashlamps were fired when the shaftencoder indicated that the platform had exactly reached thechosen position. Unfortunately, the design of the motorcontroller resulted in all commands having a timinguncertainty of up to two clock cycles, i.e. 0-4 milliseconds.Therefore an optical system was used to fire the flashlamps.A laser and photodiode were mounted on the exterior of thetank, while a piece of retro-reflective tape was attached to theplatform. When the platform reached the correct position thetape was illuminated by the laser, and the photodiode closed aswitch to fire the flashlamps. The original firing command

was modified for use as an arming command to preventinadvertent operation of the flashlamp.

D. Temperature-Sensitive Paint

Two TSP formulations were used in the present study. Thefirst consisted of Europium III thenoyltrifluoro-acetonate(EuTTA) in clear model airplane dope. This combination hadbeen used previously in air tests. It was known to be durable,waterproof, and relatively sensitive to temperature, withtypical sensitivities of around k-0.039 at room temperature.The TSP was applied on top of a base coat which consisted ofwhite model airplane dope. This paint was applied to the 30-cm scale model for initial testing. However, this paint showeda tendency to delaminate from the model surface afterprolonged soaking in water. Delamination was prevented bylimiting the amount of time the model was left in the tank.Subsequent tests of the crystal models, including the secondround of testing the 30-cm scale model, used a different paint.The second paint consisted of EuTTA in DuPontChromaClear, which is an automotive clear coat. This paintwas applied on top of a white automotive base coat. TheChromaClear-based paint proved to be much more waterresistant than the dope-based paint. No deterioration of thispaint was observed despite maintaining the models in thetank for several days. The sensitivity of both paints in waterwas less than their typical sensitivity in air. Paint calibrations(detailed in the section on experimental procedures) indicatedsensitivity coefficients of k-0.02-0.03. The reason for thelower sensitivity in water is not known. The surface of thepaint is sufficiently hydrophobic that problems wereencountered with air bubbles clinging to it. This problemwas solved by adding a surfactant wetting agent (Jet-DryDishwashing Rinse) to the water. To the authors' knowledge,this is the first use of TSP to measure temperaturedistributions in water.

IV. EXPERIMENTAL PROCEDURE

Correspondence between model and full scale cases.Once the model scale facility was constructed, data wereacquired at a variety of conditions intended to match the fullscale operating conditions shown in Table 1. These modelscale operating conditions are also shown in Table 1. Modelscale cases IM - 4M were acquired first, and were intendedto match full scale cases IF - 4F. Unfortunately, bum-outs inthe heater boards limited the amount of data which wereobtained in cases IM and 2M. There was some interest inseeing the heat transfer for a case with constant rotation, andso model scale case 5M was acquired with the platformrotating at a constant rate. The results from cases IM - 5Mwere considered sufficiently interesting to pursue furtherexperiments. After redesigning the heater boards to eliminatebum-outs, cases IM and 2M were repeated as cases 6M and7M. New cases 8M and 9M were added to match full scalecases 5F and 6F. Finally, a second constant rotation case,

10

1OM, was acquired. Constant rotation cases were run in bothclockwise and counter-clockwise directions with nodiscemable difference.Data acquisition at elevated temperature. Model scale

cases IM, 2M and 6M - IOM were conducted in water at anelevated temperature. The rationale for this procedure was totake advantage of the reduction of water viscosity withtemperature to reduce the rotation rate required in the modelscale experiments. By heating the water to a mean of 47.50 C,the maximum rotation rate required was reduced fromroughly 300 rpm to 177 rpm. High temperature dataacquisition was accomplished by first heating the water to50° C with a heating element prior to placing the model in thetank. Data acquisition was begun when the water temperaturehad cooled to 490 C, and terminated at a water temperature of46° C. Unfortunately, an incorrect calculation of waterviscosity for cases IM and 2M lead to these cases beingconducted at a slightly higher than desired rotation rate; casesIM and 2M only approximately match the Reynolds numbersof cases 1F and 2F.General data acquisition procedure. Data acquisition

typically began with the model and rotation platform beinginserted into the tank. The model was first spun throughseveral rotation cycles over the course of five to ten minutesto dislodge bubbles in the flow and to allow the model toequilibrate to the water temperature. Once the model hadbeen equilibrated it was spun through several rotation cycleswith the heater boards turned off, during which timetemperature-sensitive paint reference images were acquired.The flashlamp illumination served to freeze the modelmotion. The heater boards were then turned on, and themodel spun through three cycles to equilibrate the flow in theheat- on condition. Finally, the model was spun throughseveral cycles as TSP test images were acquired.

Model clocked+120°. Camerasees sides 1, 2,

5, & 6.

Model clocked-120°. Camerasees sides 1, 4,

5, & 8.

Positive rotation iscounter-clockwise asviewed from the top.

Model clocked

Camera sees sides 3 & 7.

Figure 13. Model is best viewed from between fins. With only onecamera, the model must be rotated (clocked) prior to running toview the desired faces. Pyramid and prism faces are numbered

separately.

Limitations on viewing angle and image acquisition. Ingeneral, it was expected that heat transfer rates would varyduring a cycle, with heat transfer being lowest when theplatform was stopped and higher when the platform wasaccelerating to its "cruise" rotation rate. Ideally imagesshould be taken at as many points during the rotation cyclesas possible. Unfortunately, as shown in Figure 5, the crystalmodel is best viewed from the sides of the tank, and indeedthe camera is set up to view the model from such a position.Since there is only one camera, images can be taken onlyover a restricted set of phase angles. In the presentexperiment, images were only taken at a 00 phase angle. Inorder to get images of all four sides of the model, theplatform was clocked +120° prior to starting a set of cycles,as shown in Figure 13. This was equivalent to rotating thecamera +120° around the spin axis of the platform.

2.5 3 5 7 9 111315171

-2.0-20~ .5 -_ _ _ _

211.0

0.5 _ _- 5 - 5

0.0-0.5 5 1t 15 20 25

-1.0

Pg -1.5 -_ _

-2.5 -__9 -1731513- 5

Time (sec)

Figure 14. Rotation rate vs time for one cycle in cases IM, 6M,and 8M. Numbered points indicate times at which the platformhas rotated 360°. Points are assigned negative numbers when

platform is rotating in negative direction.

The requirement that all images must be taken at 00 phaseangle places limitations on where in a rotation cycle data canbe acquired. Figure 14 illustrates this restriction with a plot ofplatform rotation rate vs. time, similar to Figure 2. Howeverin Figure 14, all the points at which the platform has executedan integral number of revolutions (and thus has 00 phaseangle) are marked and numbered. Starting at zero revolutions,the platform accelerates to its cruise rotation rate within threerevolutions, and continues at that rate for 16 morerevolutions. It decelerates to zero speed after slightly morethan 21 revolutions total, and then reverses direction. Thenumber of revolutions made by the platform varies with thecruise rotation rate and total period, and is different for eachcase.

Detailed data acquisition sequence. The recycle time ofthe flashlamp power supply sets an upper limit at the rate atwhich images can be acquired. Consequently, not all thedesired images can be acquired in a single cycle of theplatform. Instead, images are taken at staggered intervals overseveral cycles. For example, the complete data acquisitionprocess for case IM, 6M, or 8M, would be as follows:

1. Spin the platform through three rotation cycles withthe heater off to establish the flow in the tank. Warm

11

the flashlamps by firing them twice per cycle, tominimize intensity variations between subsequentflashes.

2. Spin the model through five rotation cycles, takingTSP reference images at selected 360° intervals butat staggered rotation values. Take images as follows:

a. Cycle 1: Rotations 0, 15, -14b. Cycle 2: Rotations 1, 16, -13c. Cycle 3: Rotations 2, 17, -12d. Cycle 4: Rotations 3, 18, -11e. Cycle 5: Rotations 4, 19, -10

3. Repeat steps 1 and 2 with heater on to acquire TSPtest images.

4. Repeat steps 1 - 3, but take images at the followingrotation values instead of the original values:

a. Cycle 1: Rotations 5, 20, -9b. Cycle 2: Rotations 6, 21, -8c. Cycle 3: Rotations 7, -21, -7d. Cycle 4: Rotations 8, -20, -6e. Cycle 5: Rotations 9, -19, -5

5. Repeat steps 1 - 3, but take images at the followingrotation values instead of the original values:

a. Cycle 1: Rotations 10, -18, -4b. Cycle 2: Rotations 11, - 17, -3c. Cycle 3: Rotations 12, -16, -2d. Cycle 4: Rotations 13, -15, -1e. Cycle 5: Rotations 14, -14, -0

6. Clock the platform +120° to view the model fromone side, and repeat steps 1 - 5.

7. Clock the platform -120° to view the model from theopposite side, and repeat step 6.

Note that one rotation point (-14) is acquired twice. Alsonote that rotation points 0 and -0 are not identical. The model"parks" briefly at this point and pauses before the beginningof the next cycle. The stagger pattern of rotation points forthe different cases is given in Table 2.

In the constant rotation rate cases (5M, 1 OM), images weretaken as follows.

1. Spin the platform through 30 rotations with theheater off to establish the flow in the tank. Warm theflashlamps by firing them every 10 rotations.

2. Continue to spin the model for another 100rotations, taking a reference TSP image every 10throtation.

3. Repeat steps 1 and 2 with the heaters on to acquirethe TSP test images.

4. Clock the platform +1200 to view the model fromone side, and repeat steps 1 - 3.

5. Clock the platform -120° to view the model fromone side, and repeat step 4.

TSP calibration procedure: A TSP calibration cell wasconstructed by placing a large water-filled beaker on a hotplate in front of the crystal growth tank, where the beakercould be viewed by the TSP camera and illuminated by theflashlamps. A 1" (2.5 cm) square TSP coupon was suspendedin the beaker, and successive images were taken as the water

temperature was increased. Temperature was measured with aplatinum RTD thermometer.

TABLE 2.ROTATION STAGGER SCHEDULES FOR DIFFERENT CASES.

1M, 6M, 2M, 7M, 3M 4M8M 9M

la 0, 15, -14 0, 19, -19 0, 4, -3 0, 6, -52a 1, 16, -13 1, 20, -13 1, 5, -2 1, 7, -43a 2, 17, -12 2, 21, -12 2, -5, -1 2, 8, -34a 3,18,-11 3,22,-11 3,-4,-05a 4, 19, -10 4, 23, -10

lb 5, 20, -9 5, 24,-9 3.-8,-22b 6,21,-8 6, 25, -8 4,-7,-13b 7, -21,-7 7,26, -7 5,-6,04b 8, -20, -6 8, 27, -65b 9, -19, -5 9, 28, -5

Ic 10,-18,-4 10,-28,-42c 11,-17,-3 11,-27,-33c 12, -16, -2 12, -26, -24c 13, -15, -1 13, -25, -15c 14, -14, -0 14, -24, -0

Id 15,-23,-42d 16, -22, -33d 17,-21,-24d 18,-20,-15d 19,-19,-0

Figure 15 shows the calibration curves obtained for cases6M-1OM. Data taken before and after these cases were runindicate that the TSP data are well-fit by Equation 22 withk-0.0281+0.0010.

1.6

1.4._

W 1.2

01

V

g 0.8

0.6

1 30 40

Temperature (deg C)

50 60

Figure 15. Coupon intensity vs. water temperature for calibrationstaken before and after cases 6M- 1 OM. Intensity normalized by

intensitv at 47.50 C.

The raw TSP data were converted to heat transfercoefficients by first taking the ratio of a heat-on image with aheat-off image taken at the same point in the rotation cycle.

12

Variations in illumination intensity from flash to flash wereassessed by examining TSP coupons which were visiblebehind the model. The coupons were attached to the outsideof the tank, with a slight stand-off to ensure they were notaffected by temperature variations within the tank. Theratioed images were rescaled to ensure that the ratio of thecoupon intensities was always equal to 1. Variations in flashintensity did not exceed 1%. The ratioed image was thenconverted to temperature by inverting Equation 22, using avalue of the sensitivity component determined from a couponcalibration as described in the previous paragraph. Theoverall model heating rate was determined by measuring thevoltage across and current through the model with a precisionvoltmeter and ammeter. The product of voltage x current,divided by the surface area of the model, gave the heatingrate. This value was divided by the TSP-derived temperatureto determine the heat transfer coefficient.

V. RESULTS AND DISCUSSION

Qualitative features of the heat transfer distribution:The variation of heat transfer rate across the model surfacefollowed roughly the same pattern in all cases. Figure 16shows the instantaneous heat transfer coefficient over side 4for each of the 42 data points taken during case IM. Thedistribution for other cases and model sides is similar. Theposition of each data point in Figure 16 with respect to therotation cycle is shown in Figure 14. At the first points in thecycle (0,1) the heat transfer distribution is quite variable, butas the rotation rate increases (2,3) so does the overall heattransfer rate on the model. Once the rotation rate reaches itspeak, a co-rotating vortex develops at the leading corner ofeach pyramid face. (The vortices are obvious in white-lightviewing of the model due to their entrainment of bubbles.)The vortex generates regions of high heat transfer at theleading edge of each pyramid face, followed by a region oflower heat transfer due to the parcel of relatively quiescentheated fluid co-rotating with the model behind each vortex.As the model continues to rotate (4-19), this region grows insize until it encompasses nearly the entire pyramid face. Theprism face is much less affected by the vortex, and does notshow a significant reduction in heat transfer rate until thelater rotations in the cycle (12-19). A small region of low heattransfer develops at the leading edge of the prism face earlyin the rotation cycle (2-4) but is largely gone by the 12throtation. The overall heat transfer rate drops significantlywhen the model begins to reverse direction (20, 21) becauseits speed drops relative to that of the fluid surrounding it.Once the model has reversed direction, the heat transferdistribution is almost the opposite of that observed in thepositive direction. The red area separating the pyramid andprism faces, which is indicative of very high heat transferrate, is an artifact. Because the 1800 turns in the serpentinetrace pattern are located at those edges of the pyramid andprism heater boards, the heat flux there is somewhat lowerthan its nominal uniform value. The surface temperature at

those board edges is thus lower than it would otherwise be,incorrectly indicating a high heat transfer coefficient.Temporally- and spatially-averaged heat transfer rates:

The time- and space-averaged heat transfer coefficient overan entire crystal face is of interest in evaluating the relativeimportance of reaction kinetics and local solute concentrationto the crystal growth process. The averaged data also allowcomparisons between cases. These serve as a check on theaccuracy of the measurements. Time- and space-averagedvalues of heat transfer coefficient are presented in Table 3.

TABLE 3(A).HEAT TRANSFER COEFFICIENTS FOR PRISM FACES AVERAGED OVER ENTIRE

VISIBLE SPACE AREA AND ROTATION CYCLE. VALUES ARE IN WATTS/CM2/C.

Case Side3 4 1 2

iM b 0.363 0.346 0.372 b2M b b 0.389 b3M 0.282 0.360 0.442 0.360 0.3154M 0.378 0.336 0.402 0.456 0.3865M 0.242 0.242 0.302 0.263 b6M(1) 0.251 0.297 0.286 0.293 0.3166M(2) 0.245 0.283 0.287 0.286 0.2837M(1) 0.273 n/a n/a n/a7M(2) 0.264 n/a n/a n/a8M(1) 0.206 0.261 0.234 0.222 0.2038M(2) 0.222 0.308 0.282 /0.245 0.2138M(3) 0.275 0.331 0.297 /0.296 0.2518M(4) 0.247 0.308 0.267 /0.256 0.2279M(1) 0.263 n/a n/a n/a9M(2) 0.331 0.386 0.345 / 0.314 0.2681OM(1) 0.135 0.162 0.149 /0.143 0.1331OM(2) 0.142 n/a n/a n/a1OM(3) 0.147 n/a n/a n/a-1OM(1) 0.140 0.171 0.150 /0.154 0.139-1OM(2) 0.144 n/a n/a n/a

Table 3 is split to present data from the prism faces (Table3a) and pyramidal faces (Table 3b) separately. Faces arenumbered as shown in Figure 13. Data are not available forall sides and all cases. In cases 1, 2, and 5, data for sidesmarked with (b) could not be obtained due to heater boardburnouts. In cases 6M-1OM, data for sides marked with (n/a)were not acquired due to time constraints. As shown inFigure 13, portions of sides 1 and 5 were viewed separatelyby the +120° and -120° camera positions. Data from the twocamera positions were averaged separately, and so thecolumns for sides 1 and 5 contain two values, one for eachcamera position. The camera which sees side 4 provides thefirst value in the column for side 1, while the camera whichsees side 2 provides the second value, and similarly for thepyramidal faces. In cases 6M-1 OM, multiple runs of the samecase are available. The separate runs are indicated by anumber in parentheses after the case number.

13

- - l l l l l l l l 1 U 'Heat06. Transfer05 Coefficient

RotationDirection 04 (W cm2 °K)

Figure 16(a). Instantaneous values of heat transfer coefficient forcase IM, sides 4 and 8. Images taken at positive rotation numberscorresponding to Figure 14.

14

15.........NotAvailable

13~41 11 v.1{d!0~~~~~~~~~~~~~~~~ M .,

9W% - 111N_1SA'X87 6'

__ _I L

5rg 4 3 2

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~...Y11..X_. .m .m.@. . .1 ..0

06 TransferRotation 05. Coefficient

Direction 04 (W/CM2 'K)

Figure 16(b). Instantaneous values of heat transfer coefficient forcase IM, sides 4 and 8. Images taken at negative rotation numberscorresponding to Figure 14.

15

TABLE 3(B).HEAT TRANSFER COEFFICIENTS FOR PYRAMID FACES AVERAGEDOVER ENTIRE VISIBLE SPACE AREA AND ROTATION CYCLE. VALUES

ARE IN WATTS/CM2/oC.

Case Side_7 8 5 6

iM b 0.286 0.298 0.324 b2M b b 0.336 b3M 0.246 0.315 0.355 0.308 0.2744M 0.308 0.292 0.315 0.360 0.3135M 0.191 0.198 0.198 0.201 0.1926M(1) 0.221 0.236 0.243 0.247 0.2396M(2) 0.214 0.227 0.241 0.238 0.2307M(1) 0.238 n/a n/a n/a7M(2) 0.236 n/a n/a n/a8M(1) 0.182 0.186 0.189 0.175 0.1848M(2) 0.192 0.226 0.219 0.190 0.1968M(3) 0.233 0.239 0.230 0.224 0.2258M(4) 0.229 0.221 0.211 0.201 0.2059M(1) 0.233 n/a n/a n/a9M(2) 0.285 0.275 0.269 0.242 0.2421OM(1) 0.128 0.130 0.135 0.127 0.1291 OM(2) 0.130 n/a n/a n/a1OM(3) 0.138 n/a n/a n/a-1OM(1) 0.133 0.138 0.134 0.139 0.135-1OM(2) 0.136 n/a n/a n/a

Heat transfer measurement variability: There isa significant degree of overlap between cases whichallows an assessment of the variability of the heattransfer measurements. Specifically,

1. Within a given case, values for sides 1 and 5generated by the two different camerapositions should not differ significantly, and

2. Symmetry dictates that sides 2 and 4 see thesame flow, and should have the same heattransfer coefficient, as will sides 6 and 8.

Table 4 shows the variation between valuesmeasured at these nominally identical conditions. Thecolumns of Table 4 show the percent variation fromthe mean values for the separate measurements ofsides 1 and 5, and for sides 2-4, and 8-6.

In general, the second set of measurements (Cases6M to 1OM) show more consistent results than thefirst set. Two exceptions to this statement are case8M(2) and the side 4-2 variation for all the 15-cmmodel cases (Cases 8M to IOM). Case 8M(2) showsa high variation which may indicate a cameraproblem while taking this case, although none wasnoted during data collection. In all the 15-cm cases,side 2 has a lower heat transfer coefficient than side4, while in all the 30-cm and 60-cm cases, sides 1, 2,and 4 have broadly comparable heat transfer

coefficients. This suggests that the side 2 data for the15-cm crystal may be bad overall, possibly due to amanufacturing defect in the side 2 heater.

TABLE 4.VARIATIONS IN AVERAGED HEAT TRANSFER COEFFICIENTS

BETWEEN FACES WHERE MEASUREMENTS ARE EXPECTED TO BEIDENTICAL. VALUES IN% EXCEPT HEAT RATE COLUMN, WHICH IS

IN WATTS/CM2.

Case ParameterHeat Side Sides Side SidesRate 1 4-2 5 8-6

iM 1.18 -7 -82M 1.193M 0.51 20 13 14 144M 0.51 -13 -14 -13 -75M 0.39 14 -26M(1) 0.92 -2 6 -2 -16M(2) 1.35 0 0 1 -17M(1) 0.937M(2) 1.458M(1) 0.99 5 25 6 18M(2) 1.03 14 36 14 148M(3) 1.05 0 27 3 68M(4) 1.03 4 30 5 89M(1) 1.019M(2) 1.04 9 36 11 131OM(1) 1.02 4 20 6 11OM(2) 1.021OM(3) 1.02-1OM(1) 1.02 -3 21 -4 2-1OM(2) 1.02

Compared to the second set of measurements, thefirst set (Cases IM through 5M) have highervariation. Not enough data exist for cases IM and2M, the first set of measurements on the 30-cmcrystal, to evaluate the level of variation in themeasurements. The variability in measurement forthe 60-cm crystal case is likely due to two factors.Due to the large crystal area, the heating rate (i.e.heat flux), and thus the surface temperatures, for thiscrystal were lower than for the 30-cm and 15-cmmodels. In addition, the paint calibration resultsindicated that the paint used for this model had lowsensitivity compared to other paint applications. Thusthe TSP brightness change between heat-on and heat-off conditions is reduced, and this degrades theaccuracy of the TSP measurement. Excluding case8M(2) and side 2 for the 15-cm crystal, the meanvariation between nominally identical sides is 6.4%,with a maximum variation of 20%.

16

TABLE 5(A).VARIATION BETWEEN HEAT TRANSFER COEFFICIENTS OF PRISM

SIDES FOR NOMINALLY IDENTICAL CASES. ONE CASE ISARBITRARILY CHOSEN AS A REFERENCE. VALUES ARE IN 0.

Case Side3 4 1 2

iM 22 21 /276M(2) -2 -5 0 / -2 -107M(2) -38M(1) -17 -15 -12 / -13 -118M(2) -10 0 6 / -4 -68M(3) 11 7 11 / 16 119M(2) 26lOM(1) -51 OM(2) 4-1OM(2) 3

For some cases repeat runs exist, and it is possibleto compare data taken for each side. For each set ofcomparable cases, the results for the first case werechosen to be the reference, and the deviation of theother cases from the reference is shown in Table 5.Note that cases IM and 6M, and 2M and 7M, weretaken at nominally identical conditions. There isgenerally good agreement between the different runsof case 6M, 7M and (for the limited data available)1OM. In particular, it should be noted that differentruns of cases 6M and 7M have significantly differentheating rates. (Heating rates are shown in Table 4.)This indicates that changes in heating rate do not biasthe measurement even though the lower heating rateused for cases 3M-5M may have increasedmeasurement variability. However there is significantvariation from run to run within cases 8 and 9. Thecause for this variation is unknown.

TABLE 5(B).VARIATION BETWEEN HEAT TRANSFER COEFFICIENTS OF PYRAMID

SIDES FOR NOMINALLY IDENTICAL CASES. ONE CASE ISARBITRARILY CHOSEN AS A REFERENCE. VALUES ARE IN %.

Case Side7 8 5 6

iM 21 23 316M(2) -3 -4 -1 -4 -47M(2) -18M(1) -21 -16 -12 -13 -108M(2) 2 8 9 / 11 108M(3) 2 8 9 / 11 109M(2) 221OM(1) -21 OM(2) 6-1OM(2) 2

Trends in averaged heat transfer coefficients:Trends in the time- and space-averaged data wereexamined by first generating an average value foreach face. Sides 2 and 4 were averaged together, aswere both measurements on side 1, to produce asingle average value for each face. Then repeat runsfor each case were averaged. (Exceptions: Case8M(2) was dropped and only side 4 data was used forcases 8M-1OM.) Figure 17 shows the heat transfercoefficients for each case sorted by side, for theprism sides only. Data for the pyramid sides aresimilar. Only the 60-cm cases show any significantdifference from side to side. The 30-cm cases showsome slight tendency towards lower heat transfercoefficient on side 3. The 15-cm cases and thecontinuous rotation cases show the least variation inheat transfer coefficient from side to side.

0.45

0.4d

0.35

D." 0.3

>0.25

0.2

0 Case 3MD Case 4MA Case 5M= Case 6MCase 8M

A Case 9Mx Case IOM+ Case -IOM

0.15

0.1

3 2+4 1Side

Figure 17. Heat transfer coefficients of prism sides sorted by side.

Since there appeared to be no significant variationin heat transfer from side to side, other trends wereevaluated by averaging results for all four sides. Theresults are shown for both the prism and pyramidfaces in Table 6, arranged by rotation rate and modelsize. In general faster rotation rate and larger modelsize result in higher heat transfer coefficients. Heattransfer coefficients on the pyramid faces are only76-91% of the values attained on the prism faces,with a mean of 84%

Space-averaged, time-resolved results: As Figure16 shows, heat transfer coefficient variessignificantly with time. The variation in spatially-averaged heat transfer coefficient for a representativecase is shown in Figure 18. Trends in heat transferwith time can be understood by referring to thevelocity time history of Figure 14. At t0O and tl13sec, the model has reversed direction and isaccelerating into flow which is still rotating in theopposite direction. Heat transfer increases rapidlywith the model rotation rate. Once the rotation ratereaches its "cruise" value, heat transfer slowly

17

decreases as the bulk of the fluid accelerates to catchup with the model. As the model decelerates, heattransfer drops rapidly because the difference betweenthe speed of the model and that of the fluidsurrounding it drops and crosses through zero.

TABLE 6(A).HEAT TRANSFER COEFFICIENTS AVERAGED OVER ALL FOUR PRISM

FACES. VALUES ARE IN WATTS/CM2/C.

Model Rotation RateSize Slow Fast Continuous15cm 0.256 0.318 0.14730 cm 0.277 0.26960 cm 0.340 0.389 0.256

TABLE 6(B).HEAT TRANSFER COEFFICIENTS AVERAGED OVER ALL FOUR

PYRAMID FACES. VALUES ARE IN WATTS/CM2/C.

Model Rotation RateSize Slow Fast Continuous15 cm 0.210 0.258 0.13330 cm 0.231 0.23760 cm 0.291 0.316 0.195

0.35

0.3

0.25

p0.2

0.15

= 0.1

* Side*Side

0.05

0

0 5 10 15

Time (sec)

Figure 18. Spatially-averaged heat transfer coe6M(1).

However, this simple picture is comtime-resolved results for the continuot19 shows time-resolved heat transfercase IOM. The heat transfer coeffisettle to a constant value. Instead it a

with a roughly 40 second time pei

shown in Figure 19 is representative -

case 10, as well as other sides of the nT

same variation. The origin of the time19 is misleading, since t0O represer

which heating was turned on, not the s

Heat-off images had been acquired o

about 75 seconds prior to the start of

model had been in continuous rotation at about 1.9revs/sec for roughly 98 seconds before the first heat-on data point was taken.

0.1

0.1

0.1

i 0.1

0.0

8

Lb__4-

L2 I

*1 - .

8

)6-

0.04

0.02

0 _0

* Side 3*Side 7

20 40

Time (sec)

60 80

Figure 19. Spatially-averaged heat transfer coefficients for case

lOM(1).

Electrical measurement of the heat transfercoefficient. The time-resolved TSP results suggestedthat some useful data might be obtained from a time-resolved, spatially-averaged method of measuring theheat transfer coefficient. In the model experiment,mean surface temperature can be determined by

monitoring the resistance of the heater boards, since

the resistance of copper varies with temperature. Toperform this experiment, the AC power supply usedto drive the heater boards for the TSP measurements

7 was replaced with a precision DC power supplyoperating in a constant-current mode, and high-precision volt and ammeters were used to measure

the voltage across and current through the model. Thewater temperature was simultaneously measured witha platinum RTD mounted a few millimeters from theinside of the tank wall at the height of the model.

vfficients for case Figure 20 verifies that there is significant time

iplicated by thevariation in the heat transfer coefficient of a crystal

is cases. Figure model in continuous rotation. The variation of heatcoefficients for

transfer coefficient has a period of approximately 36cient doents not seconds, which is consistent with the -40 secondcient does not period observed in the TSP data (i.e. 72 versus -75ippears to vary

riod. The data rev/period, respectively).The constant-speed data in Figure 20 were

riother ofeobtained using a nearly impulsive start and stop. It is

andel, show the interesting to note that the spike in heat transferaxis in Figure

its the time atexpected at the start is conspicuously absent.

lartS the hmatio. Moreover, at the stop, there is a spike in heat transfer

tart perotion. instead of the dip observed with milder deceleration.ver period of One plausible explanation is that both the initial spike

and the final dip actually do occur but are of such

18

.0, * * *

0 0

* 0

00 0

--- . 4

NI 0" I0

0 0 n

short duration that the relatively slow respondinginstrumentation does not detect them. Anotherpossibility is that the initial spike and final dip arestrictly quasi-steady phenomena and that the start andstop considered here are sufficiently rapid to disruptthe equilibrium of the boundary layers.

0.09

0.06 -

0.05

0.02

4800 5000 5200 5400 5600 5800 6000 6200 6400 6600 6800

Time (sec)

Figure 20. Time-resolved, spatially-averaged heat transfercoefficient for 15-cm model. Continuous rotation at 2 rev/sec

begins at t=5000 sec and ends at t=6500 sec.

It should also be noted that the mean heat transfercoefficient for the model, of roughly 0.065 W/cm2OK, is only about 4500 of the mean value as measuredby the TSP. This discrepancy casts doubt on theaccuracy of both sets of data and thus merits furtherinvestigation.

These results were obtained using the assumptionthat the temperature coefficient of resistance (TCR)of the heater was equal to the standard value forcopper (AR/RAT = 0.00393 0C-1). This applies tounconstrained conductors that are free to expand andcontract (in all directions) with varying temperature.In the present case, however, the thin copper film hastwo of its three axial strain components imposed bythe thermal expansion of the substrate (i.e. the muchthicker glass-epoxy circuit board and/or the PMMAcore, depending on the adhesive bond between them).The influence of substrate properties on the TCR of ametal film is a standard consideration in electricresistance strain-gage instrumentation. If the thermalexpansion coefficient of the substrate is greater thanthat of the conductor (as is possible in the presentcase), the TCR of the assembly will be higher thanthat of the unconstrained conductor.

There is also the possibility that the heaters on the15-cm model were not functioning in an idealmanner. The TSP data suggests anomalous behaviorof the heater board on side 2, as discussed previously.

Clearly, what is needed here is an in situcalibration of the resistance of this particular modelas a function of temperature. Unfortunately, this wasnot clear at the time the experiment was conductedand no such calibration was done. Moreover, at theend of the experiment, the 15-cm model was run ondirect current for long periods (full days) in anattempt to bring the entire tank to thermalequilibrium. The heater eventually failed,presumably as a result of galvanic corrosion in tapwater. (Gas bubbles were observed just prior tofailure.) However, in the data that are available,there is evidence of a substantially higher TCR. Atthe time of this writing, this analysis is incompleteand its presentation will be deferred to a futurepublication. Preliminary estimates indicate that theTCR might be higher by as much as a factor of two.If true, that would reconcile the inferred heat transfercoefficients with those obtained from the TSPmeasurements.The above-mentioned data analysis also provides

useful experimental information on the thermaltransient response of the immersed heater-substratecombination. This is an important consideration thathas received scant attention in the present paper, butwill be properly addressed in a subsequentpublication.

Conversion of heat transfer coefficients to masstransfer coefficients: Conversion factors betweenheat and mass transfer can be found using eitherEquation 19 or 20. Here, Equation 20 was used withthe Reynolds number correction factor set to unity.For this experiment the conversion factor kJh wasfound to be -6.2x10-9 m3 °K/J for the 15-cm and 30-cm model cases (SCF 700, PrM 3.7), and -3.3x10-9m3 °K/J for the 60-cm model cases (SCF2500, PrM-6.4). Applied to the TSP heat transfer data, theseconversion factors imply that mass transfercoefficients range from 8.1x10-6 m/s to 2.2x10-5 m/sfor the 15-cm and 30-cm model cases, and 6.3x10-6m/s to 1.3x10-5 m/s for the 60-cm model cases, whichare broadly consistent with observations of the fullscale experiments.

VI. CONCLUDING REMARKS

Instantaneous distributions of the heat transfercoefficient have been measured on a rotating modelin a water bath in order to estimate analogous masstransfer coefficients on a geometrically similarcrystal growing in a KDP solution under dynamicallysimilar conditions. Heat transfer measurements weremade with temperature-sensitive paint on electricallyheated models of three different sizes, each at two

19

U.U5

n n 7 A. A A. ...

w

0.04

u.u3o

different rotation rates. Two additional cases withcontinuous model rotation were studied as well. Tothe authors' knowledge, this is the first use of TSP tomeasure temperatures in water.The time-resolved TSP measurements, as well as

the TSP images, show that there is considerablepoint-to-point variation of the heat transfercoefficient over a rotation cycle. The data indicatethat the highest heat transfer coefficients occur whenthe model is accelerating through fluid moving in theopposing direction. Once the model reaches its"cruise" rotation rate, a stable vortex structure rapidly(within a few revolutions) forms and co-rotates withthe model. The vortex structure traps heated fluid andcauses a slow decline in heat transfer coefficient asthe model continues to rotate. This effect is probablymore significant on the pyramid faces since theyappear to be more directly exposed to the vorticalflow. Measurements in the continuous rotation casesindicate that if the "cruise" rotation rate was heldindefinitely the heat transfer coefficient woulddecrease to about 60% of its peak value as the vortexstructure stabilized. However in the normal course ofa rotation cycle the model rotation reverses, breakingup the vortex structure. This returns the heat transfercoefficient to its peak value, but not before reducingit sharply as the model decelerates through the speedof the fluid swirling around it.When heat transfer coefficients are averaged over

an entire face and rotation cycle, there is littlevariation from face to face. The largest variation isfound in the 60-cm scale model crystal, which wouldbe expected to have the largest degree of interactionwith the rotation platform's fins. When differentcrystal models and rotation rates are compared, theheat transfer coefficient is found to increase withmodel size and rotation rate.

It was found that continuous rotation does notresult in a constant heat transfer coefficient over themodel. Instead, the spatially-averaged heat transfercoefficient develops a stable oscillation around asteady state value, with a period roughly 72 times therevolution period. TSP images suggest that the stablevortex structure on the crystal is being periodicallydestroyed and recreated. This oscillation may berelated to an interaction between the four-foldsymmetry of the model and the three-fold symmetryof the rotation platform

Time-resolved spatially averaged heat transfercoefficients were also inferred from precisemeasurements of the heater resistance in operation.This method worked very well, but yielded valuesthat were less than half of those obtained via TSP.This discrepancy casts doubt on the accuracy of both

sets of data and thus merits further investigation. Thefindings will be reported in a future publication.The observed heat transfer response to very high

angular acceleration is qualitatively very differentfrom (almost opposite to) the response to thecomparatively mild acceleration in the model rotationcycles. At present, this phenomenon is notunderstood. Accelerations of this magnitude are notpractical for crystal growth. However, anunderstanding of these observations might lead toimproved theoretical models of relevant physicalprocesses.Mass transfer coefficients for the full scale

conditions were calculated using the Chilton-Colbumanalogy. The large ratios of the Schmidt to Prandtlnumbers and the question regarding the mostappropriate value of the Chilton-Colburn exponentgive rise to substantial uncertainty in the masstransfer predictions. Also, the broader question ofwhether the Chilton-Colburn analogy is trulyapplicable to unsteady rotating flow remainsunanswered. However, the mass transfer coefficientsthus computed were broadly in line with expectationsfrom the full-scale crystal growth experiments.

In retrospect, the authors missed an outstandingopportunity to both validate the applicability theChilton-Colbum correlation to this type of flow andto determine the appropriate value for its exponent.(Unprintable)! This would require set of runs withperfectly matched Reynolds numbers and carefullycontrolled water temperature ranging from 0°C to60°C. Over this temperature range, the Prandtlnumber varies by a factor of 4.4 (13.25 to 3.01). Alogarithmic curve fit of the data (using Equation 18with an unknown exponent) would then yield theexponent value and the goodness of fit would be anindication of applicability. Admittedly, the ratio of4.4:1 is a far cry from the Schmidt-Prandtl ratio ofnearly 400:1 that was used in computing the masstransfer coefficient for the 60-cm crystal.Nevertheless, this information would represent atremendous improvement over the present state ofknowledge. Therefore, these measurements arehighly recommended for future work.

This paper does not resolve the question of whetherNIF crystal growth is dominated by kinetics or masstransfer coefficient. If the latter is the case, however,it is worthwhile to consider how the mass transfercoefficient can be maximized. The time-resolved TSPmeasurements, as well as the TSP images, suggestsome possible strategies. Prolonged rotation at aconstant rate should be avoided, since this allows thegeneration of a stable vortex structure which reducesthe spatially-averaged mass transfer coefficient. The

20

vortex structure begins to develop within only a fewrotations of the crystal. Within about 72 rotations, thevortex structure destabilizes and reforms, but themass transfer coefficient will not regain its originalvalues. A reversal of the direction of rotation iseffective in destroying the vortex structure, butnecessarily involves an instant at which the crystal isstationary with respect to the fluid surrounding it.This causes a large transient reduction in the masstransfer coefficient which may be detrimental to thecrystal growth process.

It might be possible to destroy the vortex structurewith variations in speed, without changes indirection, relative to the surrounding fluid. Thevortex and dead-time problems might thus both beeliminated using a modified rotation schedule inwhich steady rotation is superposed on an alternatingrotation cycle (e.g. adding a vertical offset to thecurve in Figure 2). If the constant-speed offset issufficiently large and the accelerations aresufficiently small, a positive speed differential couldbe maintained between the crystal and thesurrounding fluid, at all times.The experimental methods described in this paper

could be effectively used to investigate new rotationschedules. The electrical resistance technique issimple and fast. It could be used to test a largenumber of proposed rotation schedules. The mostpromising candidates could then be subjected to thedetailed scrutiny afforded by the TSP technique.

[4] Liu T, Sullivan JP (2004) Pressure and TemperatureSensitive Paints (Experimental Fluid Mechanics Series),Springer-Verlag, Berlin.

[5] Thorsness C, Land T, Dylla-Spears R, Ehrmann P (2004)"Recent KDP Large Boul Runs - Congruent Growth"Lawrence Livermore National Laboratory MemorandumNIF0111317, Sept. 15.

[6] DeYoreo JJ, Burnham AK, Whitman PK (2002) "DevelopingKH2PO4 and KD2PO4 Crystals for the World's MostPowerful Laser" International Materials Review, 47(3), 113-152.

ACKNOWLEDGMENTS

This work was supported by the LawrenceLivermore National Laboratory under aReimbursable Space Act Agreement. The authorswould like to acknowledge the efforts of Dr. CharlesB. Thorsness and Ms. Rebecca Dylla-Spears, both ofLLNL, whose clear descriptions of the crystal growthprocess and insightful comments on this experimentaided immeasurably in the completion of both theexperimental program and this report. Also, it wasChuck's initial scientific perception (i.e. "Yes!") thatcommenced LLNL funding for this exciting project.

REFERENCES

[1] Rebecca Dylla-Spears (2004) private communication.[2] Robey HF, Maynes D (2001) "Numerical Simulation of the

Hydrodynamics and Mass Transfer in the Large Scale, RapidGrowth of KDP Crystals. Part 1: Computation of theTransient, Three-Dimensional Flow Field" Journal of CrystalGrowth, 222, 263-278.

[3] Bell JH, Schairer ET, Hand LA, Mehta RD (2001) "SurfacePressure Measurements Using Luminescent Coatings" Annu.Rev. Fluid Mech., 33, 155-206.

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