Paper No 63-HT-3

J H UENHARD Associate Professor

Deportment of Mechanical Engineering

Assoc Mem ASME

P T Y WONG Teaching Assistant Deportment

of Mechanical Engineering

Washington State University Pullman Wash

Introduction T

The Dominant Unstable Wavelength and Minimum Heat Flux During Film Boiling on a Horizontal Cylinder

Predictions of tire dominant unstable WGletength and the minimum heat flux during film boiling abote a fiat plate are found to be inapplicable in the case of boihng 011 small middotwires Nei) expressions are developed for the case of a horizontal cylinder by acshy[Ollnting for the efteet of surface tension in Ihe transverse direction upon the Taylor instabdity of fhe interface

Original measurements of WGlelengths and l1zinimum heal fluxes on small Wires are also prmgtided These data support the predictions

HE film boiling regime is eharaderized by the steady formation and release of bubbles at the liquid-over-vapor interface over a heating element Zuber and Trihus [12] proposed in 195R that th is behavior resulted from the Taylor instabili ty of the liquid-vapor interface They argued that as long as more vapor was being generated than was required to sllstain the naturalrute of growth of unstable disturhances the disturbanees would colshylapse and release hubbi es periodieaJJy It also foJJowed that the spacing between bubbles would be a c()n~lant value equal to the wavelength of the disturbance most susceptible to eollapse

Earlier studies of plane unstable interfaees by Kelvin [1 I and by BeJJman and Penningtoll [4] provided Zuber and 1ribus witmiddoth expressions for the wavelength An of the shortest unstable disturbance acnd the wavelength Ad of the most dangerous Ilnshystable disturbance Kelvin showed that aJJ disturbacnces with wavelengths less th~lI1 A where

(1)

would be stable BeJJman and Pennington showud that disshytllrbaIwes would grow most rapidly when

(2)

Tfle term IJ ill equations (l) and (2) designates the surfaGe tenshy

Numbers in braekets designate l efcrcllcc~ at the end of paper Contributed by the Heat Tranfer Diviion of THE bER1CA Soshy

C IETY OF MECHANI C ENGINEEHS for prespntation at the ASVIEshyArC hE Heat Transfer Conference and Exhibit Bo~tonIasi Augut 11 - 14 1963 Vlallllseript rece ived at A8ME Headquarters Ylarch 12 1963 Paper No 63- JlT-3

sion between a liquid and its vapm Pi and Po arc the liquid and vapor densities respeetively lt1 nel U is the gmvitational acce lemshytion

Zuber [2 I predided t he minimum hoiling heat flux with the aid of all experimental result given by Lewis [ii I Lewis found that the initially exponentiLl rate of growth of unstable disturbshytlllces predicted by Taylor [I) J applied in the range of 0 ~ 1)

~ 04 where 1) is t he ordinate of the disturbance Aceordingly Zuber fixed qmin as the lowest hent flux which genemted enough vapor to give rise to the theoretical value of the average d1)dt in this range

The minimum hoiling heat flux In in obbined in this manner is

4~ 4 712 -J J Pj - Po 60 3 Po 10 Og (PI + p)

(l)

where hj is the latent heat of vaporization 8quation (3) is suecessful in precLicting qm in for a flut plate but recent experishyments (eg [7]) have shown that qmin values for slender horizontal wires are considerably higher t han the flat plute value The present study wus thus lIlotivated by a need to know what faetors govern the failure of flat pinte themmiddotv in describing boiling upon ey linclrical heaters

Prediction of the Unstable Wavelengths The geometry of the liquid-vapor interfaee surrounding a wire

during film boiling will be assumed to take sinusoidaJJy unshyd ulat ing asymmetrical form as shown in Fig 1 The vapor hlanket surrounding the heater will be assumed to be sufficiently thin that the smaJJest radius of the interface is negl igibly larger (han the radius N of the heater The maximum perturbing amplitude a of the dominant wave wiJJ oeeur at the top of t he interface The problem of deciding whether the dominant waveshy

---Nomenclature---------------------------shya

y h

Ie k kd

]J

p

= maximum amplitude of an inlershyflwial wave

gravitational aeee leration = latent heat of vaporizatioll = wavenumber 271A = wave numbers based upon A and

Ad respeetively pressure pressures t vapor-liquid intershy

face in the vapor a nd li(1Uid phases respeetively

eflect of transverse suriace tenshysion in terms of pressure at the interface

Jp =

I =

q in =

R =

No H

r =

I

oseilbting component of P

heat flux

minimum film boiling heat flux

radius ( in partieular the radius of a heating element)

radius of a departing bubble

radius of interface in the X-1J plane

time

dislanee on axis of heater

distance upward from (he avershyage (or llndistnrbed) ordinllte of t he interface

1) = ordinate of interface A

An Ad

PI Pq

IJ

w W gtX

wavelength (in parti(ular the dominant wavelength during film boiling)

the (riticill and most danshygerous wavelengths respeeshytively

= densities of saturated liquid and satllrated vapor respeetively

surface tension between it liquid wd its V1Lpor

ungu la I frequeney = the maximum imaginary vallie

of w

Discussion on this paper will be accepted at ASME Headquarters until September 16 1963

Fig I The assumed geometry of film boiling on a horizontal cylindrical heater

x

The Contribution of Surface Tension Considered as a Sinusoidally Varying Pressure Contri bution

Fig 2 Two-dimensional model for film boiling on a horizontal cylinshydrlcol heater

length X should be equal to the cri t iea l or the lllost da ngerous wavelength will be deferred nntillater

The eflec t of surface tension along the curvature in the transshyverse direetion is the sallle as an additioNa l component of presshysure P acting to push the interface in toward the wire where

(J p = - -- (4)

radIUS

this pressurc ranges between (J I R in the va lleys and (J I( N + 0)

at the peaks of the wave it has an average value of (J ( R + ~) and an ampli t ude of (Jaj2( R + aR ) The amplitude can be approximated us (Ja 2 R since R raquo ao This pressure cont ribushyti on will act in phase with the collapsing wave and will serve to shorten the dominant wavelength

A simplified two-dimensiona l analyt ical model will be estahshylished to des(ribe this three-dimensional physical model Fig 2 shows an equivalent t wo-dimensional wave in a liquid-overshyvap or interface The contribution of curvature in the t ransshyverse direction will be included in the form of an addi t ional presshysure difference component Dp across the interface This comshyponent will be numerically equul to t he oscillating component of P

The analy tical descript ion of this model is huil t upon the work of Rayleigh a nd Kelvin (us described by Lamb [3 [) as foll ows

The interface is assumed to be disturbed by waves of the form

= a eos kx cos wt (5)

which when w is real is to be construed as t he real part of

= ae - iwt cos kx (6)

where w is the angular frequency of t he wave and the wave numshyber k is

(7 )

The liquid and vapor are assumed to be incompressible and inviscid and the perturbation is assumed to be 8Jllall in comparishyson wi th t he dep th of both fluid byerR Bern oullis equation for the system is then

The pressure difference ( Po - Pf) which balances the inertia terms on the right-hand side of equation (il ) is composed of two parts

(9)

where Dp being in phase with the disturbance is

(JDp = - a cos kc cos wt ( 10)

I 2R

and l1R the inverse radius of curva ture in the x -y pla ne is

1 0 gt = ~ = - ak cos b cos wt (11)J1 1 ux shy

Substitution of equations ( 10) and ( Il ) into equ ation ()) a nd of the resul t into equation (8) gives

(J (12 )

2R

or

(Ji (1J )

The nature of w governs the stability of t he disturbance (as Taylor [61 has shown ) If w is real the stabilizing effeet of surshyfa(e tension in the axial directi on will smooth out the disturbanee If w is imagi nary the forces of gravity and surface tension in the transverse direction will domina te and t he disturbance will inshy(reuse exponentia lly in accordance wi th equation ( 6)

In t he present (ase w pusses from real to imaginary as the right-hand side of equati on (13) passes through zero The eritishyeal wave number e is then obtained by equating the right-hnnd side to zero

(14 )

and t he critieal wavelengt h

The most dangerous wave number kd is bt ained by maximi zing ( - iw )

d(iW)] [ dk k ~ k

o ( 16)

whence

( 17)

and

( 1~)

2R2

Equations ( 15) and ( 18) show t hat as R is decreased surface tension in the transverse direction acts t o shorten the wavelength The wav elengt h will in fact diminish in direct proptlrtion to the radius whell

Transactions of the AS M E 2

Prediction of the Minimum Heat Flux A minimum heat flux predietion for t he present configurat ion

can now be formed with t he help of an energy balanee at the surface of t he heater

qm i [

latent heat] [bUbbles ] tTlt nsport per wave per hubble I oS l ill atioli

[

mininum illllllber] [Wflves per] of oSImiddotIl latlOns umt area per uni t ti me of heater

(19)

or

where wi t h aid of equation (6)

dr (21)

ell

Equation (20 ) is a direct adaptation of Zuber s min derivflshytion Three of his assumptions a re also being employed tenshytatively in t he case of cylindrical heater T hese assumptions are tested in t he subsequent sections of t his paper

The dominant wa velellgth is Ad 2 The depa rture rudius of a bubble is Ad4 1 Lewis experiment al determintion of the range in which

t he wave grows exponent ially applies in t he present conshyfigumtion

Substi tution of equations (21 ) (13 ) and ( 18) into equation (20) yields the desired prediction for qmin

v~ Pohjo [ZfJ PI - Po R PI + p

a ]h[ Y( P - Po) 1 ]-i (22) + (PI + P )fl2 - a + 2Wo

Experimental Determination of the Dominant Wavelength and qn i n

A program of experimental evaluat ions of A a nd qmin Oil slender horizontal wires has been made in order t o test the preceding predietions itesis tan(e wires of N ichrome V (00025 ~ R in ~ 00254) and Tungsten (0001 ~ R in ~ 0002) were mounted in It horizontal pyrex tube at atmospheric pressure and used t o boil isopropanol and benzene of reagent grade The apparat us shown in F ig 3 embodied the following features

~~~I~nser Variac crcJ Glass-metal seal

Cold Heater Wire PowerU Thermometer (Test Specimen) = Trans-

Py rex Tube II tarmer Nylon Fittingl r

H=~~~~~ A

Preheater Battery Motion- Picture 50V or Still Camera 20 A

(In Fron t at Tube )

Fig 3 Schematic representatlan af experimental apparatus

Journal of Heat Transfer

The t ubula r tes t section was provided wi t h a t hermometer to confirm that the liquid was saturated during runs a nd was equipped with a llOV-3A a-c resistance preheater A Nishyehrome rectangle one in wide was mounted ill the same perpenshydicular plane as the wire to facili tate measurement from photoshygraphs The lighter wires used (0001 ~ R in ~ 0010) were provided with battery-supplied direet current while the heavier wi res (00159 ~ R in ~ 00254) were fo r eonvenience supplied with a lternating current Si nce experimental results were similar for wi res of 0007J-in mdius upon which observati ons were made wit h both a -c and d-c power supplies t he effec t of current alshyternation in t he la rger wires was judged to be negligible

Both still a nd motion pictures were made of the wave action High-speed (2500 frames per sec ) motion pietures were made fo r benzene boilillg on a O07g-in radius wire a nd for isopropanol boiling on wires of 00254 00079 and 00032 ill radii Most of t he visual da ta however were obt ained from a large 111llllber of still photographs Both kinds of vislla l resul ts consisted strictly of elevation views of the wire durillg film boil illg at heat fluxes in excess of qmin

The test wires were e1eaned with soap and hot water a nd were rillsed with both hot water and acet one prior to their installation in t he t ube During the experimenta l runs care was exercised to keep t he wires from becoming red hot T his precau t ion proshytected both t he wire surface and t he liquid from deterioration The preheater was turned off immediately before observations for t wo reasons The first was to clear the Held uf vision of bubbles The second was to a void the effeet of an electric field ( reported by Bonjour et a Iii I in 1( 62 ) upon boiling The field created by the p reheater in the present experiments was foul1d to inshyereftSe qmi on t he smaller wires by a faetor of as much as fOllf

Hent flu xes were computed direetly from measurement-s of voltshyage and amperage lind the dimensions of the wire Wa velengths were measured from still photographs or from projections of the 16 mm high-speed motiol1 picture film onto a microfi lm-reader screen The probable error ill heat fluxes due to measurements was not over 3 percent in any instance The probable error in measurements of the wavelengths ranged between 1 percent for the largest one to 20 percen tmiddot for tbe sma llest

middotWhile t he experi mentH I errors were relatively low somewhat larger errors of judgment iLrose in observing t he minimum heat fluxes and wavelengths The observation of qm in was made difficult by heat conducti on into the eleetroeles which held t he wires This caused fil m boiling to collapse at the ends at heat fluxes in excess of qmin-a difficulty whieh was over(ome by watc hshying for evidence of local eollapse nway from the ends The range of uncertainty of interpretation of this observation was about plusmn 20 percent on t he average

The observation of the dominant wavelengt h was relatively straightforwa rd for wires for which R ~ 0010 in The bubble release patterns were genera lly good although both isolated deformi t ies 3nel drift in t he phase angle of oseillation along the length of the wires acted to upset the uni formity of t he patterns Fig 4( a ) shows a typical uuiform release pattern ILnd Fig 4( b ) shows typical deformities The uneertainty of in terpretashytion of wavelengt h observations in this range was usua lly about plusmn 20 percent

In the ease of wires with radii for which 0001 0 ~ R in ~ 0010 the phel1omenon of bubble merger entered to obscure t he wave pattern Bubbles generated during the first half of a eyd e tarried on the wire long enough to come into cont act with neighboring bubbles from the second ha lf of a cycle A merger would occur between the two which made ident ificati on of the 10(31 wave pat tern impossible A single instance of this beha vior is shown in F ig 4(c) for It compa rat ively la rge wire Fig 4(e) on the other hand shows a small diameter wire on which the wave admiddotion ran only be observed in u few isolated locati ons The average

The specific uncerta in t ies of each of the present data are displayed in the graphieal presentations of these data

3

Prediction of the Minimum Heat Flux A minimum heat flux predietion for t he present configurat ion

can now be formed with t he help of an energy balanee at the surface of t he heater

qm i [

latent heat] [bUbbles ] tTlt nsport per wave per hubble I oS l ill atioli

[

mininum illllllber] [Wflves per] of oSImiddotIl latlOns umt area per uni t ti me of heater

(19)

or

where wi t h aid of equation (6)

dr (21)

ell

Equation (20 ) is a direct adaptation of Zuber s min derivflshytion Three of his assumptions a re also being employed tenshytatively in t he case of cylindrical heater T hese assumptions are tested in t he subsequent sections of t his paper

The dominant wa velellgth is Ad 2 The depa rture rudius of a bubble is Ad4 1 Lewis experiment al determintion of the range in which

t he wave grows exponent ially applies in t he present conshyfigumtion

Substi tution of equations (21 ) (13 ) and ( 18) into equation (20) yields the desired prediction for qmin

v~ Pohjo [ZfJ PI - Po R PI + p

a ]h[ Y( P - Po) 1 ]-i (22) + (PI + P )fl2 - a + 2Wo

Experimental Determination of the Dominant Wavelength and qn i n

A program of experimental evaluat ions of A a nd qmin Oil slender horizontal wires has been made in order t o test the preceding predietions itesis tan(e wires of N ichrome V (00025 ~ R in ~ 00254) and Tungsten (0001 ~ R in ~ 0002) were mounted in It horizontal pyrex tube at atmospheric pressure and used t o boil isopropanol and benzene of reagent grade The apparat us shown in F ig 3 embodied the following features

~~~I~nser Variac crcJ Glass-metal seal

Cold Heater Wire PowerU Thermometer (Test Specimen) = Trans-

Py rex Tube II tarmer Nylon Fittingl r

H=~~~~~ A

Preheater Battery Motion- Picture 50V or Still Camera 20 A

(In Fron t at Tube )

Fig 3 Schematic representatlan af experimental apparatus

Journal of Heat Transfer

The t ubula r tes t section was provided wi t h a t hermometer to confirm that the liquid was saturated during runs a nd was equipped with a llOV-3A a-c resistance preheater A Nishyehrome rectangle one in wide was mounted ill the same perpenshydicular plane as the wire to facili tate measurement from photoshygraphs The lighter wires used (0001 ~ R in ~ 0010) were provided with battery-supplied direet current while the heavier wi res (00159 ~ R in ~ 00254) were fo r eonvenience supplied with a lternating current Si nce experimental results were similar for wi res of 0007J-in mdius upon which observati ons were made wit h both a -c and d-c power supplies t he effec t of current alshyternation in t he la rger wires was judged to be negligible

Both still a nd motion pictures were made of the wave action High-speed (2500 frames per sec ) motion pietures were made fo r benzene boilillg on a O07g-in radius wire a nd for isopropanol boiling on wires of 00254 00079 and 00032 ill radii Most of t he visual da ta however were obt ained from a large 111llllber of still photographs Both kinds of vislla l resul ts consisted strictly of elevation views of the wire durillg film boil illg at heat fluxes in excess of qmin

The test wires were e1eaned with soap and hot water a nd were rillsed with both hot water and acet one prior to their installation in t he t ube During the experimenta l runs care was exercised to keep t he wires from becoming red hot T his precau t ion proshytected both t he wire surface and t he liquid from deterioration The preheater was turned off immediately before observations for t wo reasons The first was to clear the Held uf vision of bubbles The second was to a void the effeet of an electric field ( reported by Bonjour et a Iii I in 1( 62 ) upon boiling The field created by the p reheater in the present experiments was foul1d to inshyereftSe qmi on t he smaller wires by a faetor of as much as fOllf

Hent flu xes were computed direetly from measurement-s of voltshyage and amperage lind the dimensions of the wire Wa velengths were measured from still photographs or from projections of the 16 mm high-speed motiol1 picture film onto a microfi lm-reader screen The probable error ill heat fluxes due to measurements was not over 3 percent in any instance The probable error in measurements of the wavelengths ranged between 1 percent for the largest one to 20 percen tmiddot for tbe sma llest

middotWhile t he experi mentH I errors were relatively low somewhat larger errors of judgment iLrose in observing t he minimum heat fluxes and wavelengths The observation of qm in was made difficult by heat conducti on into the eleetroeles which held t he wires This caused fil m boiling to collapse at the ends at heat fluxes in excess of qmin-a difficulty whieh was over(ome by watc hshying for evidence of local eollapse nway from the ends The range of uncertainty of interpretation of this observation was about plusmn 20 percent on t he average

The observation of the dominant wavelengt h was relatively straightforwa rd for wires for which R ~ 0010 in The bubble release patterns were genera lly good although both isolated deformi t ies 3nel drift in t he phase angle of oseillation along the length of the wires acted to upset the uni formity of t he patterns Fig 4( a ) shows a typical uuiform release pattern ILnd Fig 4( b ) shows typical deformities The uneertainty of in terpretashytion of wavelengt h observations in this range was usua lly about plusmn 20 percent

In the ease of wires with radii for which 0001 0 ~ R in ~ 0010 the phel1omenon of bubble merger entered to obscure t he wave pattern Bubbles generated during the first half of a eyd e tarried on the wire long enough to come into cont act with neighboring bubbles from the second ha lf of a cycle A merger would occur between the two which made ident ificati on of the 10(31 wave pat tern impossible A single instance of this beha vior is shown in F ig 4(c) for It compa rat ively la rge wire Fig 4(e) on the other hand shows a small diameter wire on which the wave admiddotion ran only be observed in u few isolated locati ons The average

The specific uncerta in t ies of each of the present data are displayed in the graphieal presentations of these data

3

0) GOOD BUBBLE DEPARTURE BUB9LES VERY NEARLY IN PHASE WITI- ONE ANOTHER (R= 0 0 254 IN q 1600 0 BTUFTZ I-1 R ISOPROPANOL)

----shy- shy- ~---middot middot0 D-CQ9 Q Q Q g nQ A 9 Q

c) GO OD BU BBLE DEPART URE PATTE RN WIT H ONE CLEAR CASE OF to4E RG ER

(R-= 00079 IN q = 56 000 a TVFT Z HR BENZENE )

b ) TYP ICAL iMPERFECT WAv E PATT ERN (R 0C2l 4 IN

q = 16000 BTUH HR ISOPROPANOL)

d RELAT IVE LY CON TI NUOUS PHASE SH IFT ALONG THE I ENGTH OF THE WIRE

( R 0 0079 rN q 54 000 BTUHZ HR ISOPROPANCL

e) EXT ENSIVE MERGI NG OF NEIGHBORING BUBBLES DOMINANT wAVE LENGTHS ARE IDE NTIFIED IN TWO PLA CES

( R =0 00 32 IN q = 118 50 0 BTUn i HR ISOPR OPANOLl

f) WAVE BEHAVIOR COMPLETELY 0 8OCUREO 8 Y 8UBBLE ERGINGS R= 00005 IN

q =450000 BTUIFT2 HR ISOPROPANOL I

Fig 4 Various pallerns of bubble doparture from horizontal wires (black objet below wire is I-in reference scale)

Table 1 Measured dominant wavelengths and heat fluxes for various wire sizes and fluids

Liquid

heater radius

dominant wave length hes t flux

i n in Bt ufthr

Isopropanol 0 0010 002 1 144 000 00015 0 029 61 000 0 0020 0 0 36 4 5 000 0 00 25 0 048 52 000 0 00 32 0 06 0 60 000 0 0050 0 112 31000 0 0063 0126 36000 0 0079 0155 17200 0 0100 0200 14300 0 0159 0288 12000 0 0201 0341 1670 0 0 0254 0 450 1640 0

Benzene 0001 5 0 0 33 112 900 0 00 20 0 048 5 0 000 0 0025 0 066 45 600 0 0040 0 068 54 000 0 0050 0 1l8 19 100 0 0079 0 173 19 700 00100 0 193 17 400

uncert ainty of wavelength observations in this range rose t o ahout plusmn 40 penmiddotent

Xo wave action could be identified on wires for whi(middoth R lt 0001 in Fig 4(f) illllstrates thi s

Observed wavelengths for the mnge of radii and for th e two liq uids are presented in T tble J The hent flux es at whieh these observations were made are induded llS well Since these heat flu xes are in eXCeSll of qm bull an additiona l set of measurements of q a nd A for given wires W HS made to (onfirm the independence of A ll nd q These da ta are given ill T ahle 2 Table3 shows the minimulll hea t Hux as It fundion of mdiu8

To additional observa tions were made to (heck the secmiddotond and third assumptiolls listed aigtove equation (22) Bubble deshypa rtnre radii R were obtained from the photographs and the ratio A H was fOrlllPd (see Table 4 ) The observations of R

Table 2 Measured dominant wavelengths for various heat fluxes for isopropanol

heat er radius heat f l ux d om i nant

wav e length

i n Btuf t 2 hr i n

C 0025 47 500 0 052 52 000 0 048 66 000 0 048 7 6 500 0046 82000 004 9 89500 0 050

I 93 60 0

103000 0 048 0 04 9

131 000 0 051 171 000 0 050

0 0100 14 300 0 200 15 500 0 194 20 100 0 197

I 22500 0 209 26 100 0 203 30 200 0 207 39 800 0 209 50 000 0 218

-

Table 3 Measured minimum heat flux for various wire sizes for isoshypropanol

heat e r (Btu ft 2hr )qm i n radius (in )

0 0025 21500 00032 22 500 0 0050 16 1 600 00063 15 00 0 ))00 12 500 0 0159 8 200

66000 0254

Table 4 Measured bubble radii and the ratio of ARI for various wire sizes for isopropanol

hea t e r radi us

bubble r adius ARb

na tur e o f wav e pat terr

in in

00063 (0 040) 0 middot 3) serious me rging 0 0079 (004 5) 0middot 5 ) s ome merging

00100 00159 0 0201 0 0254

0 055 0 0700 00900 0 1000

3 6 4 i 3 middot8 4 middot 5

11ttl e or no

merging

Table 5 Range of amplitude during which interfate grows exponentially

Liquid

hea t flux

Btu ft 2 hr

heater rad ius

in

range of exposhynential growth and uncerta in t y o f observat ion

Be nzene 56 000 0 0080 O gt Jgt 0 middot 53 ( ~10

Iso gt ro panol 16 000 0 0254 o gtJgt 0 36 ( 115)

were quite reproducible so the uncertainty of A Ribull is rough I the same as that of A The range of exponential rowth of waves was ohtained frum measurements of a few waves in ea(h of two mo tioJ] pictures as well (see T able Ii )

F or a much more detailed des( ription of the present experishyments and of da ta redu(middott ion teehniques the reader can (onsult reference 19 J

Comparison of Theory WHh Experiment Figs 5 and 6 displ ay the observed waveleng ths for isopropanol

and benzene respedively Equations (15) and (18)- the theoshyretical predictions for A and A-[lre included in these presentashytions The data indica te t ha t the dominant wavelength is genenllly only about 25 percen t higher t han the predieted Ad Furtherm ore no dominatlt wveleng th as sm all as A was ob-

Transactions of the AS M E 4

100

c

-lt c cgt c 010

b Measured Value of the-I Dominant Wave Length gt I Range of Uncertainty

of measured value ~

f Predicted A for a Harizontal Cylinder

t Ad for a Horizontal Cylinder

001 0001 001 0 1 0

Heater Radius R (in)

Fig 5 The effect 01 heater radius upon the critical and most dangerous wavelengths for iopropanol

I 00 ~-------TO---TT---------Tr-r--

Ad for a Flat Plate ~ --- - - shy

Ac for 0 Flat Plate

r

-lt c c 0 10 oMeaured Value of the-I Dominant Wave Lon Qth gt I RanQo of Uncertainty

of masured valu~ Predicted Ac for a Harizontal

Cylinder

Ad for a Horizontal Cylinder 001 LL_-L_L-~~~LL__L--L~~~~~_~

0001 001 010

Heater Radius R (in)

Fig 6 The effect of heater radiu upon the critical and most dangerous wavelength for benzene

served This suggests that the first assumption in the derivashytion of equation (22) was corred and that the present theory is successful Fig 7 is included to show that X can be evaluated

( at lilly reasonably low heat flux in excess oJ qmi The figure shows that there is no identifiable variat ion of X with q

( Fig 8 compares the present fnin data with predieted values given by equation (22) Equation (22) predicts that qmin will decrease in inverse proportion with H as H becomes large sinee the single row of bubbles will subtend au increasingly large area Actually when R becomes large additional rows of bubbles will be accommodated 3 along the top and the present description will cease to apply

While Zubers flat plate prediction does not represent the lilllit to which equation (22) tends nor docs it describe the physical circumstances of boiling on a large cylinder it is nevertheless instructive to consider it in Fig 8 It has therefore been inshycluded and with it is shown Berensons [Ill moditicntion of Zushybers predictioll The latter differs from equation (3) only in

that the constant ~ (j) (or 01 i7) has been adjusted by semishy

empirical means to 009 Berensons prediction aecordingly

Breen and Westwater [10] have observed that thi s two-dimensionshya lity is pronounced in isopropanol boiling o n a cylinder of radius R = 0336 in

bull Zuber himself [2] gave an alternate evaluation of the bubble reshylease frequency which indieated that the constant should lie between 0194 and 0255

Journal of Heat Transfer

- 008 R=00025 inc

-lt 006 c-

Ol 004 c CI)

-J 002 CI)

gt c 000 ~ a 4 8 12 16

Heat Flux q(10middotBtuft2 hr)

C Q3 R= 0010 in

-lt c 02 J f~ ~ ~ i ~ 01

~ 01 -J ~ 00 L-____L-____L-____L-____L-____L-__--I

c02 3 4 5 6

~ Heat Flux q(IOmiddotBtufthr)

Fig 7 Measured dominant wavelengths versus heat flux for isoshypropanol

The Presenf Theory for Small Cylinders (Eqn(22))

Zubers Flat Plate ~

Theory (Eqn(3))

gt CD

0 10

Present Data c

Data of LienhardE CT Iand Schrock

I Uncertainty of Berensons Semi- Empirical Observation Modificofion of Zubers

Flot Plate Theory 2 L--L-L~~~L___~__~~~~~____~~~~

a002 001 010 050 Heater Radius R (in)

Fig 8 The effect of heater radius upon qmin

fit his experimental data quite closely Equation (22) ( which is a Zuber-like prediction ) would also fit the present data closely if the rcultiplying constant were arbitrarily reduced from 7 - V3 (or 0 216) tol value of 0057 60

I t t hen rem ains to question the appropriateness of the second and third assumptions used in deriving equation (22) Table 4 shows tha t the use of Xj4 as the departure radius is entirely reasonable in the range for whicb these dttta can be obtainecl and Table 5 provides two approximate veritications of the assumed range of exponential growth

Conclusions Equation (18) preclicts dominant wavelengths during film

boiling on a horizontal wire that are 25 percent or less in exshycess of measured values It reveals a hitherto unreported deshycrease in w~welength with heater mdius resulting from t he conshyt ribution of transverse surface tension

Zuber s predict ion of the minimum heat flux on a flat plate is adapted to the present geometry and the assumptions used in this derivation a re supported wi t h experimental data The reshysulting predict ion is high by a eonstant factor as was Zubers flat plate prediction although the form of bot h predictions is apparently correct

5

100

c

-lt c cgt c 010

b Measured Value of the-I Dominant Wave Length gt I Range of Uncertainty

of measured value ~

f Predicted A for a Harizontal Cylinder

t Ad for a Horizontal Cylinder

001 0001 001 0 1 0

Heater Radius R (in)

Fig 5 The effect 01 heater radius upon the critical and most dangerous wavelengths for iopropanol

I 00 ~-------TO---TT---------Tr-r--

Ad for a Flat Plate ~ --- - - shy

Ac for 0 Flat Plate

r

-lt c c 0 10 oMeaured Value of the-I Dominant Wave Lon Qth gt I RanQo of Uncertainty

of masured valu~ Predicted Ac for a Harizontal

Cylinder

Ad for a Horizontal Cylinder 001 LL_-L_L-~~~LL__L--L~~~~~_~

0001 001 010

Heater Radius R (in)

Fig 6 The effect of heater radiu upon the critical and most dangerous wavelength for benzene

served This suggests that the first assumption in the derivashytion of equation (22) was corred and that the present theory is successful Fig 7 is included to show that X can be evaluated

( at lilly reasonably low heat flux in excess oJ qmi The figure shows that there is no identifiable variat ion of X with q

( Fig 8 compares the present fnin data with predieted values given by equation (22) Equation (22) predicts that qmin will decrease in inverse proportion with H as H becomes large sinee the single row of bubbles will subtend au increasingly large area Actually when R becomes large additional rows of bubbles will be accommodated 3 along the top and the present description will cease to apply

While Zubers flat plate prediction does not represent the lilllit to which equation (22) tends nor docs it describe the physical circumstances of boiling on a large cylinder it is nevertheless instructive to consider it in Fig 8 It has therefore been inshycluded and with it is shown Berensons [Ill moditicntion of Zushybers predictioll The latter differs from equation (3) only in

that the constant ~ (j) (or 01 i7) has been adjusted by semishy

empirical means to 009 Berensons prediction aecordingly

Breen and Westwater [10] have observed that thi s two-dimensionshya lity is pronounced in isopropanol boiling o n a cylinder of radius R = 0336 in

bull Zuber himself [2] gave an alternate evaluation of the bubble reshylease frequency which indieated that the constant should lie between 0194 and 0255

Journal of Heat Transfer

- 008 R=00025 inc

-lt 006 c-

Ol 004 c CI)

-J 002 CI)

gt c 000 ~ a 4 8 12 16

Heat Flux q(10middotBtuft2 hr)

C Q3 R= 0010 in

-lt c 02 J f~ ~ ~ i ~ 01

~ 01 -J ~ 00 L-____L-____L-____L-____L-____L-__--I

c02 3 4 5 6

~ Heat Flux q(IOmiddotBtufthr)

Fig 7 Measured dominant wavelengths versus heat flux for isoshypropanol

The Presenf Theory for Small Cylinders (Eqn(22))

Zubers Flat Plate ~

Theory (Eqn(3))

gt CD

0 10

Present Data c

Data of LienhardE CT Iand Schrock

I Uncertainty of Berensons Semi- Empirical Observation Modificofion of Zubers

Flot Plate Theory 2 L--L-L~~~L___~__~~~~~____~~~~

a002 001 010 050 Heater Radius R (in)

Fig 8 The effect of heater radius upon qmin

fit his experimental data quite closely Equation (22) ( which is a Zuber-like prediction ) would also fit the present data closely if the rcultiplying constant were arbitrarily reduced from 7 - V3 (or 0 216) tol value of 0057 60

I t t hen rem ains to question the appropriateness of the second and third assumptions used in deriving equation (22) Table 4 shows tha t the use of Xj4 as the departure radius is entirely reasonable in the range for whicb these dttta can be obtainecl and Table 5 provides two approximate veritications of the assumed range of exponential growth

Conclusions Equation (18) preclicts dominant wavelengths during film

boiling on a horizontal wire that are 25 percent or less in exshycess of measured values It reveals a hitherto unreported deshycrease in w~welength with heater mdius resulting from t he conshyt ribution of transverse surface tension

Zuber s predict ion of the minimum heat flux on a flat plate is adapted to the present geometry and the assumptions used in this derivation a re supported wi t h experimental data The reshysulting predict ion is high by a eonstant factor as was Zubers flat plate prediction although the form of bot h predictions is apparently correct

5

References N Zuber and M Trihus FUJmiddotther Remarks on the Stabilitv

of Boiling Heat Transfer UCLA Rept No 58-5 Univ of Calif t Los Angeles J anuary 1958

2 N Zuber Hydrodynamic Aspects of Boiling Heat Transfer Atomic Energy Commiss ion Report No AECU-4439 Physics Rnd Mathematics June 1959

3 H Lamb H ydrodynamics 6th edition Dover Publications New York N Y 1945 p 455 et seq

4 H Bellman and R H Pennington Effects of Surface T ension and Viscosity on Taylor Instability Quar Appl Math vol 12 1954 p 15

5 D J Lewis The Instability of Liquid Surfaces When Accelershyated in a Direction Perpendicular to Their Planes II Proe Roy Soc (London) Series A-202 1950 p 81

6 G I Taylor The Instability of Liquid Surfaces When Acshycelerated in a Direction Perpendicular to Their Planes I Proc Roy Soc (London) Series A-20 I 1950 p 192

7 J H Lienhard and V E Schrock The Effect of Pressure Geometry and t he Equation of State Upon the Peak and Minimum Boiling Heat Flux ASME Paper No (J2- - HT-3 AIChE-ASME 5th Nat1 Heat Transfer Conference Houston T exas (August 5 to 8 1962)

8 E Bonjour 1 Verdier and L Weil Improvement of Heat Exshychanges in Boiling Liquids Lnder the Influence of an Electric Field AIChE Preprint 7 AIChE-AS1E 5th Nat[ Heat Transfer Confershyence Houston Texas (August 5 to 8 1962)

9 P T Y Wong Effects of H eate r Geometry Upon the LiqllidshyVapor Configuration in Film Boiling Wash State rniv In~t of Tech Bull No 26i Fehruary 1963

10 B P Breen and 1 W Westwater Effect of Diameter of Horshyizontal Tubes on Film Boiling Heat Transfer Chem Enol PTOg vol 58 no 7 July 1962 p 67

11 P J Berenson Transition Boiling Heat Transfer From a Horizontal Surface MIT Heat Transfer Laboratory T echnical Rept No 17 1960

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