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E D I T O R I A L S T A F FE d i to r , J . J . JA K L IT S CH , JR .

Product ion Ed i to r ,S TE LLA RO B INS O NEdi tor ia l Prod. Asst . ,

K I R S T E N D A H LH E A T T R A N S F E R D I V I S I O N

Ch a i rman , F . W. S CHMIDTS ecre ta r y , C . J . CRE ME RSl ior Techn ica l Ed i to r , E . M. SPARROWTechnica l Ed i to r , W. A U N G

Techn ica l Ed i to r , B . T. CHAOTech n i ca l E d i to r , D . K . E DWA RDS

Tech n i ca l E d i to r , R . E ICHHO RNTechnica l Ed i to r , P . GRIFFITH

Techn ica l Ed i to r , J . S . LEETechn ica l Ed i to r , R. SIEGEL

P O L I C Y B O A R D ,C O M M U N I C A T I O N S

Ch a i rman and V i ce -P res iden tI . B E R M A N

Members -a t - La rgeJ. W . L O C K EJ . E . O RTLO FF

M. J . RA B INSW . J . W A R R E N

Pol icy Board Representa t ivesBasic Eng ineer ing, F. LANDIS

Genera l Eng ineer ing, D. D. ACKERIndustry, M. M. L IVINGSTON

Power , R. E. REDEFResearch , G. P. COOPEFCodes and Stds. , P . M. BRISTEF

Compu te r Tech no logy Com. ,A . A . S E IRE G

Nom. C o m . Rep. ,S . P . RO G A CK IBusiness Sta f f

345 E. 47th St .New York, N. Y. 10017

(212 )644 -7783Mng. Dir . , Pub l . , C. 0 . SANDE RSON

O F F I C E R S O F T H E A S M EPresident , 0 . L . LEWIS

Exec. Di r . & Sec'y, ROGERS B. FINCHTreasu re r , RO B E RT A . B E NN E TT

EDITED and PUBLISHED quarterly at theoffices of The American Society ofMechanical Engineers, United EngineeringCenter, 345 E. 47th St.. New York, N. Y.10017. Cable address. "Mechaneer."New York. Second-class postage paidat New York, N. Y., and at additionalmailing offices.CHANGES OF ADDRESS must be received atSociety headquarters seven weeks beforethey are to be effective. Please sendold label and new address.PRICES: To members. $25.00, annually; tononmembers, $50.00. Single copies, $15.00 each.Add $1.50 for postage to countries outside theUnited States and Canada.STATEMENT from By-Laws. The Society shall notbe responsible for statements or opinionsadvanced in papers or . . . printed in itspublications (B 13, Par. 4).COPYRIGHT 1978 by the American Society ofMechanical Engineers. Reprints from thispublication may be made on conditions that fullcredit be given the TRANSACTIONS OF THEASME, SERIES CJOURNAL OF HEATTRANSFER, and the author and date of

publication stated.INDEXED by the Engineering Index, Inc.

transactions of the nSIHEPublished Quarterly byThe American Society ofMechanical EngineersVolume 100 Number 3AUGUST 1978

journal ofheattransfer387

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The Effect of Internal Solidification on Turbulent Flow Heat Transfer and Pressure Drop in aHorizontal TubeS. B. Thomason, J. C. Mulligan, and J. EverhartEffect of Subcooling on Cylindrical MeltingE. M. Sparrow, S. Ramadhyani, and S. V. PatankarTurbulent Flow of Water in a Tube with Circum ferential ly Nonuniform Heating, with or withoutBuoyancy

R. R. Schmidt and E. M. SparrowFree Convection across Incl ined Air Layers with One Surface V-CorrugatedS. M. ElSherbiny, K. G. T. Hollands, and G. D. RaithbyCorrelation Equations for Turbulent Thermal Convection in a Horizontal Fluid Layer HeatedInternally and from Below

F. B. CheungTransient Free Convention from a Suddenly Heated Horizontal Wire

J. R. Parsons, Jr. and J. C. MulliganAn Experimental Investigation of Heat Transfer and Buoyancy Induced Transition from LaminarForced Convection to Turbulent Free Convection over a Horizontal Isothermally Heated Plate

H. Imura, R. R. Gilpin, and K. C. ChengPartial Spectral Expansions for Problems in Thermal ConvectionE. J. Shaughnessy, J. Custer, and R. W. DouglassMeasurement of Buoyant Jet Entrapment from Single and Multiple Source s (77-HT-43)

L. R. Davis, M. A. Shirazi, and D. L. SlegelConvection in a Porous Medium Heated from Below : The Effect of Temperature DependentViscosity and Thermal Expansion Coefficient (77-HT-56)R. N. Home and M. J. O'SullivanBubble Growth in Variable Pressure FieldsO. C. Jones, Jr. and N. ZuberExperimental Study on Bubble Nucleation in the Oscillating Pressure Field (77-HT-xx)K. Hi j ikata, Y. Mori, and T. NagataniMagnetic Field Effects on Bubble Growth in Boiling Liquid MetalsC. P. C. Wong, G. C. Vliet, and P. S. SchmidtEarly Response of Hot Water to Sudden Release from High Pressure

J. H. Lienhard, Md. Alamgir, and M. TrelaTwo-Dimensional Multiple Scattering: Comparison of Theory with Experiment (77-HT-48)D. C. Look, H. F. Nelson, A. L. Crosbie, and R. L. DoughertyMulti-Dimensional Radiative Transfer in Nongray GasesGeneral Formulation and the BulkRadiative Exchange Approximation (77-HT-51)S. S. Tsai and S. H. Chan

492 Corresponding States Correlations of the Extreme Liquid Superheat and Vapor Subcooling (77-HT-

496

503

508

514

520

527

20) J. H. Lienhard and A. H. KarimiLaminar Boundary Layer Transfer over Rotating Bodies in Forced FlowMin-Hsium Lee, D. R Jong, and K. J. DeWittDetermination of Unknown C oefficients in Parabolic Operators from O verspecified Ini t ial-BoundaryData

J. R. Cannon and P. C. DuChateauPrediction of Temperature Profiles in Fluid Bed Boilers (77-HT-66)

J. L. Hodges, R. C. Hoke, and R. BertrandPerformance Ranking of Plate-Fin Heat Exchanger Surfaces (76-WA/HT-31)

J. G. Soland, W. M. Mack, Jr., and W. M. RohsenowAn Experimental and Analytic Study of a Unique Wet/Dry Surface for Cool ing Towers

J. M. Bentley, T. K. Snyder, L. R. Glicksman, and W. M. RohsenowHeat Transfer During Piston Compression

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531 Aerothermochemlstry of Metal Erosion by Hot Reactive Gases (77-HT-12)A. Gany. L. H. Caveny. and M. Summerfield

TECHNICAL NOTES537 Effect of Clrcumferenllal Wall Heat Conduction on Boundary Conditions for Heat Transfer In aCircular Tube

J. W. Baughn539 Heat Transfer In the Entrance Region of a Straight Channel: Laminar Flow wtth Uniform Wall

TemperatureM. S. Bhatll and C. W. Savery

542 Mixed Convection about a Sphere with Uniform Surface Heat FluxA. Mucoglu and T. S. Chen

544 Radiation Augmented Fires within EnclosuresA. T. Modak and M. K. Mathews

547 Approximate Radiation Shape between Two SpheresJ. D. Felske

549 Tube Wall Temperatures of an Eccen trica lly Located Horizontal Tube within a Narrow AnnulusR. W. Alperl

552 A Finite Element Thermal Analysis Procedure for Several Temperature Dependent ParametersE. A. Thornton and A. R. Wieting

554 Temperatures In an Anisotropic Sheet Containing an tnsulated Elliptical HoleM. H. Sadd and I. Mlsktegtu

556 Heat Conduction In Axisymmetric Body-Duct ConfigurationT. Mlloh

559 Closed form Solutions for Certain Heat Conduction ProblemsA. K. Naghdl

DISCUSSION561 Discussion of a previously published paper by

A. A. Sfelr561 Discussion of a previously published paper by

A. F. Emery and F. B. GessenerANNOUNCEMENTS

385 Call for Papers-18th ASME/AIChE National Heat Transfer Conference394 International Conference on Numerical Methods In Thermal Problems564 Information for Authors

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S BB ThomasonGraduate Research Assistant-

J . C. Mull iganProfessor.

Mem. ASME

J . EverhartGraduate Research Assistant.

Department of Mechanical andAerospace Engineering,

North Carol ina State University,Raleigh, N. C. 27650

The .Effect of Internal Solidificationon Turbulent Flow Heat Transferand Pressure Drop in a Horizontal

ubeA simple analysis of the steady-state heat transfer and pressure drop in turbulent flow ina tube is presented for the case involving a "thin," steady -state frozen deposit on the inside tube wall. Sparrow-H allman-Siegel type interna l flow convective heat transfer expressions and Blasius type pressure drop expressions are employed while neglecting secon d order interface curvature effects. Experimental heat transfer and pressure drop dataare presented for comparison. It is shown that simple a nalyses of the type developed canbe used to predict heat transfer and pressure drop in tube flow under freezing conditionsand that, for the experimental conditions tested, basic agreement between theory and experiment was obtained. It is also shown experimentally that small nonuniform ities in walltemperature can produce wide variations in pressure drop when a frozen layer ex istswithin a tube.

In t roduct ionMelting and freezing problem s commonly occur in a wide varietyof processes and situations which involve fluid low and heat transfer.Fundamentally, these phase change problems have in common acharacteristic nonlinearity which complicates their analysis and whichrenders each problem somewhat unique. In addition, sometimescomplicated and oftentimes perplexing physical phenomena occurin the melting-freezing process which invalidate conventional analyses. A variety of studies dealing with the analytical as well as experimental aspects of particular problems have appeared in the literature over the past decade. Good summaries of these have beenpublished by Muehlbauer and Sunderland [1], Boley [2], Bankoff [3],and Gilpin [4], A survey of some of the more recent literature was

carried out by Shamsunder and Sparrow [5].Those melting-freezing problems involving internal flow oftenpresent unusually troublesome modeling and analytical difficulties,especially when natur al convection and supercooling are significantsuperimposed mecha nisms. In some cases of supercooling, it has beenshown [4] that an annu lar phase differentiation does not even existbut, instead, a dendritic matrix occludes the flow area. The impor-

Contributed by the Heat Transfer Division for publication in the JOURNALOF HEAT TRANSFER. Manuscript received by the Heat Transfer Division July22,1977.

tance of these internal problems was recognized as early as 1916 whenBrush [6] pointed out some general freezing ch aracteristics of watermains. It has only been in recent years, however, that they have received sub stantia l deve lopment. In cases of relatively large flow ra teand liquid phase superheat, wherein an annular growth might beexpected for most liquids, most of the work has been directed to th elaminar flow case. Some early one-dimensional analyses applicableto annular freezing in laminar tube flow were presented by Londonand Seban [7], Foots [8], and Hirshburg [9]/ In 1968 Zerkle andSunderland [10] published an important analytical and experimentalinvestigation of therm al entranc e region annular freezing in laminartube flow, wherein the steady-state problem was reduced to theclassical Graetz problem. E xperimental studies of freezing in thermalentrance region laminar flow were also recently carried out by Depewand Zenter [11] and Mulligan and Jones [1 2]. DesRuisseaux andZerkle [13] used the results of [10] to show how one can predict theoverall system conditions which will lead to the blockage of a laminarflow in a tube with a simple entrance and entrance length and witha prescribed static pressure at the exit. They also considered theprocedures which would be involved in the prediction of blockageconditions for other upstream flow systems. The transient flowcharacteristics preceding the blockage, however, were not considered.

As indicated, most of the work on internal flow with annularfreezing has addressed the freezing process in laminar flow. However,many intern al flow problem s involve fluid flow which is turbulent innature and, specifically, little exists in the literature to indicate the

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effects of turbul ent f low on freezing in tubes. The m aterial present edhere represents the results of a s tudy of the inte rnal an nular freezingin the thermal entrance region of a cooled horizontal tube when theflow is s teady, hydrodynamically fully developed and tu rbulen t. Th edevelopment of an approximate analysis which is re latively s impleand dire ct is outl ined, a long with the results of i ts use in pre dictingthe s teady-s ta te hea t t rans fe r and p ressu re d rop . E xper imen ta l hea ttransfer and pressure drop data are presented for comparison.A n a l y s i s

The inlet f low is assumed to be of uniform temperature, greaterthan the freezing temperature, and to be hydrodynamically fullydeveloped at the entrance where the cooling begins. The tube walltemperature is taken to be uniform, constant, and lower than thefreezing temperature of the liquid. The bulk temperature of the liquiddecreases as it flows down the tube, promoting solidification and thuscausing the thickness of the frozen shell to increase with dis tance d ownthe tub e. This change in flow area results in an acceleration of the fluidand produces a two-dim ensional velocity distr ib ution . Th e steady -state interface profile exists primarily as a result of the l iquid phas esuperhea t a t the en t rance . Wi thou t th i s supe rhea t , the tube wou ldfreeze solid. A sketch of the process is shown in Fig. 1.

For purposes of the theoretical analysis , the following init ia l assumpt ions a re made :

1 The flow is s teady, axisymme tric , fully developed hydrod ynamica lly , tu rbu len t , and o f un i fo rm temp era tu re a t the en t rance tothe cooling section of the tub e.

2 Th e fluid is incompressible and Newto nian, the propertie s ofboth phases are constant, and their densit ies assumed equal andeva lua ted a t sa tu ra ted l iqu id cond i t ions . T he f lu id P ra nd t l num beris assumed to be in the intermediate range and evaluated at the tubeinlet conditions.

3 Rad iatio n, free convection, viscous dissip ation , and body forcesare negligible . Also, i t is assumed that the f luid acceleration is notsuffic iently s trong to influence the s tructure of the turbulent f low.

4 The tem pera ture at the l iquid-solid interface is equal to thefreezing temperature.

5 Th e wall tem pera ture is uniform, consta nt, and below thefreezing temperature of the l iquid.

6 The tube wall has negligible therm al resis tance.With these a ssumpt ions , the s tandard equa t ions govern ing con-

Fig. 1 Illustration of freezing section entrance and steady-state frozen shellprofile

servation of mass (continuity) and l inear momentum for the l iquidphase, the heat conduction in the solid phase, and the convective heattransfer in the interior l iquid phase can be formulated and appliedin the usual way. The interfacial radial he at transfer coupling can beexpressed as

, i>T,\ I-k s ~ qe\ = 0dr \r=i \r=iT he boundary cond i t ions a t r = 0 for z > 0 are

dr dr

(1 )

(2)and a t r = &(z) and z > 0

u2 = u r = 0 , T = Ts = T fAt the tube wall , r = rw and z > 0

TS = TW< T fa n d a t t h e en t r a n c e , 2 = 0 a n d 0 < r < rw

T( = To = constant, vr = 0vz = Fully developed turbulent f low

velocity profileIt was shown by Zerkle and Sund erlan d [10] that the growth of an

(3)

(4)

(5a)(5b)

-^Nomenclature-.c = specific heat of the liquidD = inside diameter of tubeg = acceleration of gravityGz = Graetz number, Re-Pr-D/Lh = mean heat transfer coeffic ient based on

arithmetic mean of inlet and exit , bulkt e m p e r a t u r e s , Q/irDL[(TM + Tbe)l2 -T f]hz = local convective heat transfer coefficient, q ifl(Tb - T f)hp = total pressure head, Pip + u 2 2 / 2hp = radially averaged total pressure head

h* = nondimensional radially averaged totalpressure drop in test section, (h po hp)-z*/PBAll = difference in upstream and downstreamreservoir liquid levels

k = the rmal conduc t iv i tyL = length of test sectionm = mass f low rateNu = mean Nusse l t number based on tube

d iamete r , hD/keN u 2 = local Nusselt number, hz(2b)lkgNu = fully developed asymptotic Nusselt

n u m b e rP = static pressure

PB = pressure drop based on mean inlet velocity and diameter and calculated usingBlasius fr ic tion relationship

Pr = P rand t l number , viaP* = nondimensional test section static

pressure drop, (Po _ P e ) / ( p V 2 / 2 )q = heat fluxqif = interface heat fluxq* = nondimensional ra te of heat transfer ,

(T bQ-T be)l{TM~T,)Q = to tal ra te of heat transferr = radial coordinateRer> = inlet Reynolds num ber based on tube

d iamete r , VD/vR e 2s = local Reynolds number based on thediam eter of the ice interface, vz {2b)/nT = t e m p e r a t u r eTb = bu lk l iqu id tempera tu reTb = mean bu lk tempera tu re , (T b0 +

Tbe)l2Tf = f reez ing tempera tu reTm* = nond imens iona l wa l l t empera tu rep a r a m e t e r , ks(T / - Tw)lk((Tb0 - Tf )v = velocityvz = radial mean of axial velocity compon e n t

V = mean inlet velocity at z < 02 = axial posit ion coordinatez* = nondimensional axial posit ion, 4z/(D-

R e o - P r )a = thermal diffusivity of the l iquid5 = radiu s of the solid phase interface5* = nondimensional solid phase interface

rad ius , 8/rwv = kinematic viscosity6* = nondimensional l iquid phase tempera

tu re , defined in equation (14a)p = f luid densityrrz = axial component of l iquid shear s tressTif = fluid shear stress at solid-liquid inter

faceS u b s c r i p t se = evaluated at tube exit conditions = refers to the l iquid phase0 = refers to conditions a t the freezing section

in le tr = component in the radial directions = refers to the solid phasew = evaluated at the tube wallz = refers to axial co mpo nent as well as des

ignating axial posit ion dependence

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ice shell initiated in a flowing liquid in a tube is smooth and gradual.Since the LI D of a tub e is relatively large, the change in the thick nessof the shell with re spect to chang e in axial posit ion is small , tha t isoildz is of a small magnitude. It can also be shown by expansion anduse of the chain rule on the governing conservation equations that thesecond derivatives with respect to axial posit ion are of the order ofd25/oz2 and (d


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In terms of nondimensional variables, the expression for the heattransfer rate becomesJo

dz* (25)In (5*)1/T *Thus, once the profile of the ice shell (& * versus 2*) is determined, theheat transfer rate is evaluated directly from equation (25) by numerical integration.A n a l y t i c a l R e s u l t sThe system of governing equations, with approximations, wassolved using a digital computer and the results a re presented in Figs.2 ,3 ,4, and 5 for selected values of the various independent param eters. The analytical results showed only slight dependence on P randtlnumber over the intermediate range considered (Pr = 8,10,12) an dthus, data are presented for only one value of Pr, chosen equal to 1 0.0so that comparisons between theory and the experiments utilizingwater would be meaningful.The theoretical dimensionless heat transfer rate, q* , and thequantity (d*)1/Tw* are plotted versus dimensionless axial position 2*in Fig. 2 for various turbulent flow cases (differing only in Reynoldsnumber) as well as for the lam inar flow case as taken from Zerkle andSund erland [10]. It can be seen th at for a given value of 2* the turbulent flow dimensionless heat transfer rate q* is very sensitive to

changes in R eynolds number over the range of values considered, thissensitivity diminishing somewhat at the higher Reynolds numbersAn increase in R eynolds number while holding 2* constant results inan increase in q* , and for all cases plotted the turbu lent flow dimensionless heat transfer rates are seen to be significantly higher thanthose for the laminar flow case for comparable values of 2* . If o n econsiders flows of the same Reynolds n umber in tubes with varyingL/D, bearing in mind tha t 2* varies directly with LID, it can be seenfrom Fig. 2 tha t q* increases with an increase in LID. Thu s a longertube yields greater heat trans fer, as one might expect, than does ashorter tube with flow of the same inlet Reynolds number.Another useful comparison may be made by examining flows ofdifferent Reynolds numbe rs in tubes of the same LID. For this caseremembering that 2* varies inversely with changes in Re D, Fig. 2shows that an increase in inlet Reynolds number yields a lower valueof q* for a given tube . This of course does not indicate a decrease inactual heat transfer, but simply that the ratio of actual heat transferto maximum possible heat transfer has decreased. The actual heattransfer from t he fluid is increased due to the higher values of the heattransfer coefficient resulting from higher inlet, and thus local,Reynolds numbers. However, due to the increase in flowarea resulting

1000

1.00.80.60.40.200.001

f" \- * / > * & >

0---1XL-__J I L J _ L i _ I I I 1 I 10.01 0.1

15000

I 1 ) 1 ) 1 1 1

F i g . 2 Va r i a t i o n o f (h e d i me n s i o n l e ss h e a l t r a n s fe r a n d so l i d p h a se I n te r f a cera d i u s w i t h a x i a l p o s i t i o n f o r P r = 1 0 , Tw ' = 3 , a n d va r i o u s Re yn o l d s n u m b e r s

F i g . 4 Va r i a t i o n of d i me n s i o n l e ss p re ssu re d ro p w i t h a x i a l p o s i t i o n f o r P r -1 0 a n d v a r i o u s R e y n o l d s n u m b e r s a n d w a l l t e m p e r a t u r e s

I00O

I00

0.00I

T =0

I.OF i g . 3 Va r i a t i o n o f t h e d i me n s i o n l e ss h e a t t r a n s fe r a n d so l i d p h a se I n te r f a ce F i g . 5 Va r i a t i o n of d i me n s i o n l e ss p re ssu re d ro p w i t h a x i a l p o s i t i o n f o r P r =ra d i u s w i t h a x i a l p o s i t i o n fo r P r = 1 0 , R e D = 5 0 0 0 , a n d v a r i o u s w a l l t e m p e r - 1 0 , R e 0 = 5 0 0 0 , a n d v a r io u s w a l l t e m p e r a t u r e s .a t u r e s .

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from thinnin g of the ice shell, the local Reynolds num bers and thusthe heat transfer rates do not increase in proportion with increasesjn inlet Reynolds num ber. Thu s, the ratio defining q* decreases wthincreasing flow rate.Fig. 2 also shows (6*)1/T > * for the turbu lent flow case to be sensitiveto Reynolds num ber over the range of 2* values where the degree oftube closure is significant. In this range the effect of increasing Rep,for the case of a constant L/D, is that of increasing (5*)1/T are also minimal. Fig. 2 also shows(jt-ji/Tu,* t 0 be greater for the turbulent flow cases than for the laminarcase with a comparable value of L/D, indicating a reduction in the icethickness in going from the lam inar flow case to the tur bule nt case.

Fig. 3 is a plot of q* and (5*)1/T "* versus 2*, bu t for cases involvingvariations of the dimensionless wall temperature parameter Tw*. Itmay be seen from thisfigure hat , for a given value of 2*, the turbulentflow dimensionless heat transfer rate q* is only moderately sensitiveto changes in Tw* and thus wall temperature, with increases in Tw*resulting in higher values of


8/185

The entrance and test section were made continuous and were construc ted of thin wall copper tubing, 1 .45 cm in inside diameter. Theconstant temperature bath and inlet reservoir provided suffic ientcapacity and control to allow the sett ing of inlet tempe ratur e with nonoticeable f luctuation. Th e entran ce section of the tube as well as theexternal side of the test section were well insulated and the circulatingwater was f i l tered to eliminate algae and particulates . The L/ D forthe entrance and the test section were 118 and 80.1 , respectively.

A counterflow, forced circulation cooling system using m ethan olas coolant was employed to control the tube wall temperature bycirculating the methanol in the cooling jacket of the test section. Acons tan t tempera tu re ba th was used to con t ro l and ma in ta in thecoolant temperature, and this was done with no noticeable fluctuation.Methanol was used as a coolant because of its low viscosity at lowtempera tu res , and a lk hp pu mp was used to c ircu la te the me thano lat a flow rate sufficient to insure a negligible tempe ratu re difference(usually on the order of 0.2C) between methanol entering and leavingthe test section. Great care was taken in the design of the test sectionand in the conduct of the experiments to insure that the nonunifor-mity in tube wall tempe ratu re was minim ized. In most of the experimentation this variation was less than 0 .8C.

A therm ocoup le at location 1 insured no variation in inlet watertemperature. Thermocouples were installed in grooves in the tube wallat each end an d the m iddle of the test section (at locations 2 , 3 , and4 in Fig. 6) to accurately measure the wall temperature. T he exit m eantemperature was measured by a thermocouple located in a smallmixing cup a t the exit of the t ube ( location 7 in Fig. 6) . Th e m ixingcup was designed to tra p and mix fluid issuing from the tub e yet allowthe exit reservoir to impose the prescribed exit s ta tic pressure. Thetest section inlet pressure was measured using a diaphragm-typestrain gauge transducer that was calibrated to 0 .03 cm of water. Thesystem was leveled with a surveying transit so that a ll points a longthe axis of the entrance and test section were at the same elevationwithin 0 .04 cm. The head height of the constant head reservoirs a tinlet and exit was measured to within 0 .03 cm with a cathetometer.All thermocouples were calibrated with NBS certif ied thermomet e r s .

The inlet water temperature was held constant a t 5 .6C and theexit s ta tic pressure at 7 .9 cm of water in all the experim ents reportedhere. The water level in the upstream reservoir and the coolant temperature were varied to produce different s teady-state conditions.Once the desired upstream water level and coolant tempe ratur e wereatta ine d, coolant c irculation through t he test section was begun, thuslowering the tube wall temperature below 0C. If the coolant tempera ture was suffic iently low, the water s ta rted freezing spon taneously. However, there were certain combinations of coolant temperature and upstream water level a t which freezing did not beginwhen th e coolant f low was started. For these cases, i t was necessaryto temporarily block the tube at the exit of the test section, thus

stopping the water f low and init ia t ing the freezing process. Oncefreezing started, the tube exit was unblocked and flow allowed to resum e. Solidif ication then continu ed until the s teady -state conditioncorresponding to the chosen coolant temperature and upstream reservoir level was atta ined. To insure th at a true s tead y state had beenreache d, most e xperim ents were allowed to run for a t least one hourafter equilibrium conditions were initia l ly atta ined. The steady-statemass f low rate was then determined from water samples t imed andweighed to within approximately two percent.

The dimensionless heat transfer ra te q* was evaluated directly asa tem perat ure ratio , and the dimensionless axial posit ion (z* = 4/Gz)was evaluated from the inlet Reynolds number Rec, Prandtl numberPr, and test section LID. All fluid properties were evaluated at thearithmetic average of the inlet and exit bulk temperatures. TheNusselt number, based on the ari thmetic mean of inlet and exittemperatures instead of the more usual logarithmic mean, was computed from

Nu = 2


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0.5

O CONSTANT L/D = 80.1

J L6.0 6.2 6.4 6.6 6.8 7.0Fig. 7 Compa rison of experim ental and ana lytical heat transfer data.

6.0 6.2 6.4 6.6 6.8 7.0Fig. 8 Compa rison of experim ental and analy tical pressure drop data.

num ber, the the oretical results are also plotted in Fig. 7 for the caseof flow in a tube with constant L/D, that value of L/ D being chosenthe same as that of the actual tube used in the experimental investigation. While the magnitude of the experimental values differsomewhat from the analytical data , i t should be noted tha t the s lopesof the analytical constant LI D l ines agree with the s lope of the experimental results within the s ta tis t ical s ignificance of the data .

Experimental and analytical dimensionless pressure drop data areplotted versus dimensionless posit ion in Fig. 8 . This f igure shows ascattering of the experimental P* data in the near neighborhood ofvalues predicted by the analysis . The experimental da ta were obtainedfor Tw* values of approximately 3.1 with a deviation of approximately0.25 for Reynolds numbers in the range of 4000 to 5000. I t can beseen in Fig. 8 th at in general the mean of the scattered expe rime ntalP* values exceeds the analytical predictions by approximately 20percent. Freezing in a tube with a nonuniform wall temperature distribution would likely cause greater pressure drop than would freezingin a tube with a uniform wall temperature equal to the mean. Examination of the tabu lated data shows that, in general , the experim entalP* data with the greatest deviation from the analytical results arethose having Tw* values near the high end of the range considered,and i t was observed during the conduct of the exp eriments th at runswith higher Tw* values ( lower actual wall temperatures) displayedwall t empera tu re d is t r ibu t ions w i th the g rea te s t deg ree of nonu n i -

formity. Thus, the scatter and posit ive shift of the experimental P*data is believed to be due primarily to the non-uniformities in walltemperature which experimentally result from increases in Tw*.Th e experimen ts were designed to provide data in an in terme diaterange of z *. During the conduct of the experiments , the z * range waslimited on the low side (0 .006) by troublesome supercooling effectsand by intolerable wall temperature variations on the high side(0.007). Although th is appears to be a rathe r narrow and arbit raryrange, it does repre sent a realistic situati on with a 5* of approx imate ly0 .85. Moreover, if the data confirm the analysis a t these arbitraryconditions within reason, then i t seems likely that the analysis willbe satisfactory at other conditions which do not obviously violate theassumptions of the analysis . The actual applicabil i ty of the theory,however, a t much larger and smaller values of z* remain s to be tested .

During the experiments every attempt was made to observe the iceinside the tube. All that could be determined was that the ice shell waswell formed with a distinct interface. In view of the appreciable liquidphase superh eat and the high Reynolds number s, i t seems reasonableto assume the ice interface was smooth.C o n c l u s i o n s

The analysis was found to be simple, direct, and acceptably accuratein representing steady-state conditions. Heat transfer predictionswithin 20% of the data me asured e xperim entally were obtained forall conditions of tube wall temperature and Reynolds number. In fact,i t is believed that the source of this deviation, a lbeit nominal, wasprimarily error in the measurement of the l iquid exit bulk temperature . An inconsistency of 0 .4C in this measurem ent is a ll that wouldbe required to produce the 20% deviation unde r the conditions of theexperiments . Thus, i t is believed that the analysis predicts actualconditions much more accurately than indicated by this comparison.

Analytical predictions of the pressure drop across the tube agreedvery well with the experim ental da ta for comparable conditions wh enthe imposed wall temperature was uniform. However, conditions ofhigh Reynolds num ber and /or very low wall tem pera ture tend ed toproduce actual pressure drops in excess of the prediction, a resultbelieved to be caused by the more severe nonuniformity of wall temperature characteris t ic of these conditions. In fact , i t was found experimentally that the actual pressure drop was extremely sensit iveto th i s nonun i fo rmi ty , wh i le on ly modera te ly dependen t upon theactual wall temperature as predicted by the analysis . Conclusionspertainin g to the sensit ivity of pressure drop to wall temp eratur e forfreezing in tube f low have bee n repor ted previously [10, 11 , 12] , a lthough it has not been demonstrated previously that th is is actuallya sensit ivity to the nonuniformity in the wall tempera ture rather thanto i ts mean value.

One very im porta nt conclusion of the analytical work is tha t a verypeculiar c rust profile occurs during freezing in turbu lent flow in tubes.A concave curvature was found to occur in the z* midrange, a lthoughthis result could not be verif ied experimentally and must await corroboration in experiments designed specifically to accomplish this .A c k n o w l e d g m e n t

The authors wish to acknowledge the f inancial support of this resea rch by the Na t iona l Sc ience Founda t io n th rough NS F Gran t G K3 8 1 3 1 .R e f e r e n c e s

1 Muehlbauer, J. C. and Sunderland, J. E., "Heat Conduction withFreezing and Melting," Applied Mechanics Reviews, Vol. 18, 1965, pp. 951 -959.2 Boley, B. A., "The Analysis of Problems of Heat Conduction and Melting,High Temperature Structures and Materials," Proceedings of 3rd Symposiumon Naval Structural Mechanics, Pergamon Press, Oxford, 1963, pp. 200-315.3 Bankoff, S. B., "H eat Conduction or Diffusion with Change of Phase,"Advances in Chemical Engineering, Vol. 5, Academic Press, New York,1964.4 Gilpin, R. R., "The Effects of Dendritic Ice Formation in Water Pipes,"International Journal of Heat Mass Transfer, Vol. 20,1977.5 Sham sundar, N . and Sparrow, E. M., "Analysis of MultidimensionalJou rnal of H eat Transfer AUGUST 1978, VOL. 100 / 393

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Conduction Phase Change Via the Enthalpy Model," ASME JOURNAL OFH E A T T R A N S F E R , Vol. 97, Aug. 1975, p. 333.6 Brush, W. W., "Freezing of Water in Subaqueous Mains Laid in SaltWater and in Mains and Services Laid on Land," Journal of American WaterWorks, Vol. 3,1916 , pp. 962-980.7 London, A. L. and Seban, R. A., "Rate of Ice Formation," Trans. ASME,Vol. 65,1943, pp . 771-779.8 Poots , G., G., "On the Application of Integral Methods to the Solutionof Problems Involving the Solidification ofLiquids Initially at the FusionTemperature," International Journal of Heat Mass Transfer, Vol. 5,1962 , pp.525-531.9 Hirschbu rg, H. G., "Freezing of Piping Systems," Kaltetechnik, Vol.14,1962, pp. 314-321.10 Zerkle, R. D. and Sund erland, J. E., "Th e Effect of Liquid Solidificationin a Tube Upon the Laminar-Flow Heat Transfer and Pressure Drop, ASMEJ O U R N A L O F H E A T T R A N S F E R , Vol. 90,1968, pp. 183-190.11 Depew, C. A. and Zenter, R. C, "Laminar Flow Heat Transfer andPressure Drop w ith Freezing at the Wall," International Journal of Heat Mass

Transfer, Vol. 12,196 9, pp. 1710-1714.12 Mulligan, J. C. and Jones, D. D., "Experim ents on Heat Transfer andPressure D rop in a H orizontal Tube with Internal Solidification," InternationalJournal of Heat Mass Transfer, Vol. 19,1976, pp. 213 -219.13 DesRuisseaux, N. and Zerkle, R. D., "Freezing in Hydraulic Systems "ASME Paper No. 68-HT-24,1968.14 Sparrow, E. M., Hallman, T. M., and Siegel, R., "Turbulent HeatTransfer in the Th ermal Entrance Region of a Pipe with a Uniform Heat Flux "Applied Science Reviews, Vol. 7,1959, pp. 3 7-52.15 Kays, W. M., "He at Transfer: Turbule nt Flow Inside Smooth Tubes,"Convective Heat and Mass Transfer, McGraw-Hill, New York, 1966, pp 173187.16 Shibani, Ali A. and Ozisik, M. N., "Freezing of Liquids in Turbulent FlowInside Tubes," to appear in Canadian Journal of Chemical Engineering, (Nowin press).17 Shibani, Ali A. and Ozisik, M. N " A Solution of Freezing of Liquids ofLow Prandtl Number in Turbulent Flow Between Parallel Plates," ASMEJ O U R N A L O F H E A T T R A N S F E R , Vol. 99,1977, pp. 20-24.

INTERNATIONAL CONFERENCE ON NUMERICALMETHODS INTHERMAL PROBLEMSSw ansea, United Kingdom, July 2 -6 , 1979The Conference will be concerned with the application of numerical methods to linear and non-linear thermal problems,

with the main aim of establishing the sta te of the art of such applications. Analysis of industria l and technological applications will be especially welcomed. It is expected that the published proceedings will form a definit ive volume onthe subject .

Abs tracts are invited on the topics outl ined in the provisional progra m below. Other pape rs of merit in related topicswill a lso be considered for inclusion. Th e abstra cts should be approxim ately 30 0 words in length and submi tted beforeJ a n u a r y 1,1979. Final papers should be submitted by April 1,1979. All paper s will be published in the conference proceedings.

Provisional Program1 . Hea t Conduc t ion2. Phase Change3 . Free and Forced Convection4. Coupled Conduction and Convection5. T u r b u l e n t H e a t T r a n sf e r6. H eat and Mass Transfer in Porou s Bodies7. Thermal and Drying Stresses8 . Geo the rmal P rob lems9 . Mathemat ica l and Computa t iona l T echn iques

10 . Industria l and Scientif ic ApplicationsPersons wishing further details of submitt ing abstracts should contact:

Dr. K. MorganDep ar tm en t o f Civ i l E ng inee r ingUniversity CollegeSwansea , SA2 8PPUni ted Kingdom

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E. M. SparrowS. Ramadhyan iS. V PatankarDepartment of Mechanical Engineering,

University of Minnesota,Minneapolis, Minn.

Effect of Subcooling on CylindricalMeltingAn analysis is made of the melting of a subcooled solid surrounding a heated circular cyl-inder. The solution is facilitated by coordinate transformations w hich immo bilized boththe mooing interface betw een the melt region and the solid and the moving temperaturewave that diffuses into the solid. The actual solutions were carried out numerically viaa finite-difference procedure which circumvents the nonlinearity associated with themoving interface. Results were obtained for a wide range of a subcooling parameter andof the Stefan number. It w as found that the subcooling can have a marked effect on themelting characteristics. Depending on the degree of subcooling, the surface heat transfercan be several times greater than that for no subcooling. Furthermore, at high levels ofsubcooling and at long melting times, the liquid layer thickness may be only a small fraction of that without subcooling. Subcooling also sharpens the differences between cylindrical and plane melting. The ratio of the heat flux for cylindrical melting to that forplane melting increases substantially due to subcooling, while the thickness of the cylindrical melt layer is only a fraction of that of a corresponding plane melt layer.

I n t r o d u c t i o nHe at conduction problems involving solid-liquid phase change have

generated an extensive analytical literature. In the main, theseanalyses have been concerned w ith transien ts in which, initially, thephase change medium is at its melting temperature. It appears thatonly in the case of the melting (or freezing) of a plane layer has theproblem of an initially subcooled (superheated) medium been examined. This is the well-known Neumann problem which is describedin various heat conduction texts.The preoccupation of the literature with situations without initialsubcooling (superheating) is remarkable in that such a condition isdifficult to achieve in practice, especially when the dimensions of thephase change medium are large. For instance, in m elting experim entsperformed in our laboratory, painstaking control and long waitingperiods (~ one week) were required to attain tem perature uniformityjust below the m elting point in a 0.3 m (1 ft) cubical solid before theactual phase change studies could be initiated. Various degrees of

subcooling (superheating) can be expected in practice, for othergeometrical configurations as well as for the plane layer.The present paper is concerned with phase change in a subcooled(superheated) medium surrounding a circular cylinder. For con-creteness, the analysis and the prese ntation of results will be couchedin terms of melting, but th e same a nalysis and results apply to freezingwith only slight rep hrasing. Specific consideration is given to a cylinder situated in a solid phase-change material whose initial tem-

Contributed by the Heat Transfer Division for publication in the JOURNALOF HEAT TRANSFER. Manuscript received by the Heat Transfer DivisionDecember 9,1977.

perature T is lower than the melting temperature T* . A t time equalszero, the temperature of the cylinder surface is raised to a value Twand maintained constant thereafter.An annular melt layer forms around the cylinder, and the outerradius of the melt region grows with time. In addition, since thetemperature of the liquid-solid interface (i.e., the melting tem peratureT*) exceeds that of the solid, a temperature wave emanating from theinterface m oves into the solid. Thus, the re are, in effect, two movingboundaries in the present problemthe liquid-solid interface andthe forward edge of the temperature wave.E ven w ithout subcooling, the cylindrical melting problem does notyield an analytical solution, and this same sta te of affairs p revails withsubcooling. A finite difference technique was employed here in conjunction with a formulation in which both the liquid-solid interfaceand the forward edge of the tempe rature wave were immobilized (i.e.,are rendered stationary in a transformed coordinate system). Theenergy equations for the liquid and the solid which result from thedual immobilization are discretized by an implicit difference scheme,

while an explicit representation is used for the interfacial energybalance. This treatment enables the solution to march steadily forward in time, without iterations being required at each time step.The problem involves two prescribable parameters. One of theseis the Stefan number (based on (T w - T*)), to which numerical valuesof 0.1, 0.5, and 1.0 are assigned. The second parameter is the subcooling ratio (T* - T)/(TW T* ) which ranged between zero andfour.The presentation and discussion of results will be focused on threequantities: the instantaneous rate of surface heat transfer, the time-integrated surface heat transfer, and the instantaneous position ofthe liquid-solid interface. These results will be examined from twoviewpoints. In one, the subcooling-affected results are compared with

Journal of Heat Transfer AUGUST 1978, VOL. 100 / 395Copyright 1978 by ASME

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those without subcooling. In the other, the cylinder and plane layerresults are com pared at a given degree of subcooling.

A searc h of the li tera tur e failed to reveal any prior work on the effects of subcooling in a cylindrical melting arrangement. For phasechange without subcooling in this geometry, Stephan and Holzknecht[ l ] 1 have reported results for the t imewise movement of the l iquid-solid interface, but do not report heat transfer results . Various investi gator s (e.g., [2, 3] and th ose cited in [4]) have use d c oord inatetransforma tions to immobilize a moving phase-change boun dary, bu tthe prese nt dual immobilization appea rs not to have been previouslyemployed. Furthermore, the prior use of phase boundary immobilization appears to have been numerically implemented with either animplicit or an explicit difference scheme. A s already noted , the pre sentschem e involves the selective use of implicit an d explicit differencingand is free of the i terations tha t are typically encou ntered in the implicit scheme and of the highly restr ic tive t ime step l imitations of theexplicit scheme.A nalysi s

A schema tic diagram of the subcooled melting problem is prese ntedin Fig. 1. Th e diagram shows dimensional an d coordinate designationsand the pa r t ic ipa t ing tempera tu res T T* , and T*,. The insta ntaneous radii of the liquid-solid interface and the forward edge of thesolid-phase temperature wave are r* and r , respectively. A s will beelaborated la ter , r is chosen large enough so as not to affect thesurface heat transfer or the interface posit ion. Subcooled solid(temper ature T ) occupies the region r > r. T he the rma l p rope r t ie sof the liquid and solid phases are assumed to be equal in order to keepthe number of prescribable parameters within reason.

Go vern ing E qua t ion s . T he hea t trans fer p rocesses in the l iqu idand solid regions are governed by the he at equa tion w rit ten in cylindrical coordinates

5Hdt r dr \ dr I' ,s (1 )The oth er key equation is the energy balance at th e l iquid-solid interface, in which the heat conducted to the interface from the l iquidis equated to the la tent heat required by the phase change plus theconduction into the solid .

-k(bTe/dr) = PX(d5 f/dt) - k(dTJdr) at r = r*(t) (2 )The other thermal boundary conditions to be satisfied by the solutiona re tha t

T=TW a t rm T = T* a t r*(t), T=T at r(t) (3 )In addition, prior to the onset of melting, the temperature of the

Fig. 1 Schem atic diagram of the sub cooled melting problem

phase-chan ge material is uniform and equal to T (< T* ).Since the problem does not permit an exact analytical solution,

numerical techniques w ere adopted because they can provide resultsof assured accuracy (in contradistinction to approximate methodssuch as the Heat Balance Int egral) . To facil i ta te solutions via f initedifferences, i t appears advantageous to work in a domain of unchanging size in which the grid can be fixed once and for all. This requires that the moving boundaries be immobilized and, to this end,a transformatio n of variables is carried out. The transform ation alsoserves to introduce dimensionless variables and parameters .

First , new space coordinates -q and , respective ly for the liquid a ndthe solid regions, are defined as

(r - rw)/8e, ( = (r - r*)/Ss (4)Since rw < r < r* in the l iquid and r*


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Finally, dimensionless temperature variables for the liquid and solidregions are introduced as

Bt = {Te-T*)/(T a-T*), 6S = (T* - T.)I(TW - T) (7)along with the subcooling ratio

= (T* - T)HTW - T*) (8)

ddr

dr_ 1

0.So l ut i o n M e t ho do l o g y . A s has been not ed earlier, solutions willbe obtained via num erical tech niques . A key feature in the implementation of the solution method is the selection of A s, which represents a posit ion in the solid that l ies beyond the region affected bythe temperature wave emanating from the interface. One approachwould be to take A s as a constan t tha t is large enough to insure th atthe temperature wave never reaches A s during the entire range of rvalues to be used in the com putatio ns. A lthough attractiv e becauseof its simplicity, this approach leads to an inefficient use of the finite-difference grid . This is because, in the solid , the grid spans theentire space between the interface and A s at a ll t imes, while a t re latively short tim es only the region adjacent to the interface is ther mallyactive. Thu s, the grid poin ts lying outside the therm ally active zonedo not contribute to the progress of the solution even though they areinvolved in the computations.

A mo re effective and equally s imple ap proach is to le tAs = C A P (16)

where C is a constant for each case. With this model, A s increasesalong with the timewise increase of the melt layer thickness A( . T h u s ,the porti on of the solid which is spa nne d by the finite-difference gridStows larger with t ime. Inasmuch as the thermally active zone also

enlarges with t i me, this m odel enables relatively effic ient use of theentire grid . I t should be noted, however, that A s/A ^ is not s tr ic tly aconstant in cylindrical systems and so, to make the method work, Cis chosen to accom modate the largest value of A s/A ^ occurring duringthe computa t ion pe r iod .

The C values varied from case to case, with a typical magnitudebeing about 100. Verif ication run s were made with other values of Cto insure that the results for the surface heat transfer and the interfaceposit ion were insensit ive to the selected C values to within 0 .1 percen t .

The transformed conservation equations (11) and (12) were recastas implicit f inite-difference equat ions. To solve the result ing difference equa t ions for the tem pera tu re d is t r ibu t ions 6e and 0S at t ime r= r


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2 -

_

----

1 * = = "

r PLANEt~n x x c

Ste0.51.0

Ste =LAYER

c r r r

0.1

-

J 1

_ _ .

_ L _

^^"~~~

^ ^

_ .

rzzzr^Lzsz.

i i i i

_._----

. --

i i

- - " " '

T * -Tw

. - .

^^

~ L J _ - L _

4 _

To, -- T * -

" " 2 _

ZZ 1-.z=0.5-"""4

_ - 2__ |

i i i

q_

- 2

10" 10" 10" 10^ I0 1 IOdTFig. 2 Com parison of the instantaneous surface heat fluxes with and withoutsubcooling

from two viewpoints. In the first, the subcooling-affected results arecompared with those without subcooling. In the second, the resultsfor cylindrical melting are compared w ith those for plane m elting atthe sam e degree of subcooling.Subcoo ling/No-subc ooling Com parison. The comparison ofresults w ith and w ithout subcooling is made in F igs. 2, 3, and 4, forthe instantaneous surface heat flux, the time-integrated surface heattransfer, and the interface position respectively. These figures willbe discussed successively.The ratio of the instantaneous surface heat fluxes w ith and w ithoutsubcooling, q and g0 respectively, are plotted in Fig. 2 as a functionof the dimensionless time variable r. Results for Ste = 0.1 are presented in the lower graph, while the upper graph is for Ste = 0.5 and1.0. The curves are parameterized by the temp erature ratio (T* -

T*,)I(TW - T*), which is a measure of the degree of subcooling.The curves appearing in these graphs are for cylindrical melting.In addition, short horizontal segments have been inserted along theleft-hand margin of the figure to denote the g/ g 0 results for meltingof a plane layer. Since, as will be demonstrated shortly, both q an dgo for a plane layer vary as T" 1/2 , then g/go is independent of r.Therefore, the horizontal line segments could have been extendedacross the entire graph but w ere not so drawn in order to avoid confusion.The deviations of the curves of Fig. 2 from an ordinate value of 1.0give an immediate indication of the effect of subcooling on the instantaneous heat flux. As expected, subcooling increases the surfacehea t flux (i.e., g/go ^ 1), and the extent of the increase is more markedat higher degrees of subcooling. In this regard, it m ay be noted tha tfor subcooling ratios up to a bout 0.5, the hea t flux is not very muchaffected by the subcooling (i.e., g/g 0 ~ 1.1 to 1.25, depending on Ste).On the other hand, for subcooling ratios on the order of four, thesubcooling becomes a major factor in establishing the level of the h eatflux.

It may further be observed tha t at small subcooling, g/go does notvary appreciably w ith time. In co ntrast, at large subcooling, there isa substantial timewise increase in g/go.The Stefan numbe r has only a small effect on the re sults at smallsubcooling ratios, but becomes a significant parameter when thesubcooling is large. In gene ral, higher values of g/go occur at largerStefan num bers.A t small values of the time T, the heat flux ratios for cylindricalmelting approach those for the plane layer. When the subcooling issmall, the g/qo values for the cylindrical case do not deviate significantly from those for the plane layer for the entire range of times thatwas investigated. On the other hand, at large subcooling, the timewiseincrease of g/go for cylindrical melting gives rise to significant deviations from that of the plane layer. In general, the instantaneoushea t flux for cylindrical melting is more affected by subcooling than, is that for plane-layer melting, with the plane case serving as the lower'bound for the g/go of the corresponding cylindrical case.

The increase in the surface heat flux due to subcooling results fromthe involvement of the mass of the solid as a heat sink that is additional to the laten t heat required by the melting process. The relativecapacity of this sensible he at sink is greater at large Stefan numbers,with a corresponding increase in the surface he at flux. In the case ofcylindrical m elting, the fanning out of the radial he at flow lines in thesolid gives rise to a thermal resistance that is lower than that for aplane layer. In ad dition, the fanning out enables more mass to becomeinvolved with the energy storage function. Both these factors contribute to the greater increase in surface heat flux encountered incylindrical melting than in plane melting. These factors are especiallyeffective when the subcooling is large and w hen the cylindrical geometry asserts itself, i.e., for thicker melt layers.Before leaving Fig. 2, it is approp riate, for com pleteness, to statethe equation for the instantaneous surface h eat flux for a plane layer.From [5], Chapter 5,

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2 -

Ste0.5

r-TW-

^~ ~

TcD_ T *

' "2 -

Lr . - jm0 . 5 -

Ste = 0.

PLANE LAYER

10'-3

0.Qo

Fig. 3 Comparison of the t ime-Integrated surface heat transfers with andwithout sub cooling

Fig. 4 Comparison of the Instantaneous liquid layer thicknesses with andwithout subcooling

qp = k(Tw-T*)/(Trat) l'2erft (20) transcendental equationwhere f is a constant whose value is obtaine d by solving the following Ste 1 - erf\p (21)

Journal of Heat Transfer AUGUST 1978, VOL. 100 / 399Downloaded 21 Dec 2010 to 193.140.21.150. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cf
http://lr.-jm0.5-/http://lr.-jm0.5-/


16/185

If i/'o den ote s the solution of equati on (21) for (T* - T ) = 0 (nosubcooling), then


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4

- 5

- 4

- 3

-

- 2--

f= r T = r f ^ *

_ *

"~i "T " i

S te0.10 .51.0

^ * * s

l i

gjf

I

/ / ' ~ 4/ / "/ f; 4-0.5/ M'N/ f # /' / ' ' / / / /

w///i/'/"I / //.'r "// / / A / 1/'[of/ ///X\4'2" / / // / N

S&v / / A /

^ " 4 T w - T *i i i i i i i i10" 10 2 10" 10"

TFig. 5 Co mparison of the instantaneous surface heat fluxes for cylindricaland plane melting

10'

Fig. 8 Comparison of the time-integraied surface heat transfers for cylindricaland plane melting

slowly than the thickness of a plane layer.Of particular n ote is the ma rked effect of subcooling on the thick

ness ratio . A t a high degree of subcooling and at long melting t im es,the thickness of a cylindrical melt layer may be only a third of t ha t

of a plane m elt layer.Numerical values of 0c can be obtained from an equation such as(25) , with q replaced by &(. Figs. 4 and 7 are used for (6V^o)c a n d

( < W ^ P )o , and &tpo is from (24).

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i.O

0.5

Ste = 1 " \ > ^ " v ^ " " " " -

^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ - ^ ^ ^ ^ ^ /S te = 0.5 v " -^^^~\~^^^^~^^\T^^~\~^~~~~^^

Ste = 0.1 ^ \ ^ \ .

i i i i i i I i i i i i i i r -

Too-r~ - 0~ - 0 . 5 -^ 1^ 2

- 4 -=-=0""^ 0.5^ 1^ 2^ i 4 , i

1.0

8 / p0.5

10" 10" I0 UFig. 7 Comparison of the instantaneous liquid layer thicknesses for cylindricaland plane melting

Concluding RemarksThe results of this analysis have shown that subcooling of the solid

can have a mark ed effect on melting due to an emb edde d cylindricalheat source. Both the instantaneo us and t ime -integrated surface heattransfer are substantia lly augmented (relative to the nonsubcoolingcase) when the subcooling ratio exceeds two. The extent of the augmen tation is greater at larger times as the cylindrical geometr y asse rtsitself. In the presence of subcooling, the m elt layer is th in ner th an i twould be without subcooling, and this effect is accentuated as themelting p roceeds. A t high levels of subcooling and at long m eltingtimes, the layer thickness may be only about a f if th of that for nosubcooling.

Subcooling also sharpens the differences between cylindrical andplane melting. The ratio of the surface heat fluxes for cylindrical andplane melting, which always exceeds unity , increases substantia llydue to subcooling. Similarly , in the presence of subcooling, thethickness of a cylindrical melt layer is only a fraction of the thicknessof a corresponding plane melt layer.

In the absence of other published results for cylindrical melting ofa subcooled solid , the only possible comparisons between the presentresults and t he l i tera ture are for subcooled plane melting and for cylindrical melting without subcooling. Verification runs comparing thepresent numerical solution with the classical Neumann (analytical)

solution for subcooling of a plane melt layer yielded agreement towithin 0 .1 percent or better . For cylindrical melting without subcooling, the only information published in the literature is for the m eltlayer thickness (i.e., surface heat transfer results are not given), withthe m ost accurate available results being those obtained numericallyby Steph an and H olzknec ht [1] . The graphical prese ntatio n of theinformatio n in [1] does not enable a very precise reading but, w ithinthis l imitation, good agreement with the present results appears toprevail .

R efe ren ces1 Stephan, K ., and Holzknecht, B., "He at Conduction in Solidification ofGeometrically Simple Bodies," W arme-und Stoffiibertragung, Vol. 7,1974, pp.200-207.2 Saitoh, T., "Numerical Method for Multi-Dimensional Freezing Problemsin A rbitrary Domains," ASME JOURNAL OF HEAT TRANSFER, Vol. 100, No.2, 1978, pp. 249-299.3 Duda, J. L., Malone, M. F., Notter, R. H., and Vrentas, J. S., "A nalysisof Two-Dimensional Diffusion-Controlled Moving Boundary Prob lems," International Journal of Heat and Mass Transfer, Vol. 18,1975, pp. 901-910.4 Bankoff, S. G., "He at C onduction or Diffusion with Change of Phase,"in Advances in Chemical Engineering, Vol. 5, pp. 75-150, A cademic Press, NewYork, 1964.5 E ckert, E . R. G., and Drake, R. M., Analysis of Heat and Mass Transfer,McGraw-H ill, New York, 1972.

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FL R. SchmidtE. M. SparrowDepartment of Mechanical Engineering,

University of Minnesota,Minneapolis, Minnesota

Turbulent Flow of Water in a Tubewith Circumferentially NonuniformHeating, with- or without BuoyancyExperiments have been performed to study the effect of circumferentially nonuniformheating o n the fully developed turbulent heat transfer characteristics for water flow ina horizontal circular tube. The use of a specially fabricated tube enabled heat to be supplied to the fluid over half its circumference, while the other half was unheated. Separatesets of experiments were conducted with the heated portion at the top and at the bottom.By varying the temperature level, the Prandtl number was varied from 3.5 to 11.5. TheReynolds number ranged from 3,000 to 70,000. The measurements enabled circumferential distributions of the Nusselt number, wall temperature, and heat flux to be determined, and circumferential average Nusselt numbers were also evaluated. Both the circumferential average and circumferential local results demonstrate that significantbuoyancy effects are present for bottom heating at low Reynolds numbers and high Ray-leigh numbers, and a criterion is deduced for the onset of these effects. The top heatingexperiments were not affected by buoyancy. The buoyancy-unaffefted circumferentialaverage Nusselt numbers increase smoothly over the entire range of Reynolds numbers,and the Prandtl number dependence is correlated as PrQA1 . These Nusselt num bers arewithin about ten percent of literature correlations (applicable for Re > 10,000) for circumferentially uniform thermal conditions. The circumferential distributions of the Nusseltnumber and temperature on the heated wall tend toward uniformity when the turbulenceis well developed (i.e., higher Reynolds numbers) or when buoyancy is present. The temperatures on the unheated wall are generally lower than the bulk temperature in the absence of buoyancy, but, when buoyancy is active, they are above the bulk temperature.

I n t r o d u c t i o nTurbulent heat transfer in circular tubes has been the subject of

extensive investigation for ma ny years. In the main, the work has beenconcerned with the case where the wall heat flux and wall temperatureare uniform around the circumference of the tube. Although circumferential uniformity is an attractive simplification in both analysisand experiment, i t does not reflect reali ty in many established applications (e .g . , boiler , condenser, and heat exchanger tubes) as wellas in newer technologies.

Wi th regard to the la tte r , the receiver tubes of solar collectors area case in point. The solar flux arriving at a tubular receiver will, ingeneral , vary around the circumference of the tube. The exte nt of thevariation depends on whether the solar collector is concentrating ornonconcentrating and, if concentrating, on the degree of concentrationand the sharpness of the focus. In addition, for horizontal or inclinedtubes, another issue that m ay be relevant is whether th e upper portion

Contributed by the H eat Transfer Division for publication in the JOURNALOF HEAT TRANSFER. Manuscript received by the Heat Transfer DivisionSeptember 23 , 1977.

of the circumference is more or less s trongly heated than the lowerportion. The orientation of the heating may be a factor in the creationor suppression of buoyancy-induced secondary motions superposedon the forced convection mainflow.

The foregoing considerations provided motivation and directionfor the present research. Experiments were performed for turbulentflow in a horizontal c ircular tube which was uniformly heated overhalf of i ts c ircumference and was unheated over the other half. T opermit examination of possible buoyancy effects , Reynolds numbersas low as 3 ,000 were investigated. Furthermore, two orientations ofthe heated section were examined: ( i) heating above and adiabaticbelow (top heating), ( i i) heating below and adiabatic above (bottomheati ng). To investigate the buoyancy-unaffected regime, the rangeof the Re ynolds n um ber w as extended to abou t 70,000. Although asingle working fluid, water, was employed in the experiments, its bulkPrandtl number was systematically varied to cover a relatively widerange, 3 .5 to 11.5. Wall- to- bulk tem pera ture differences were keptsuffic iently small so that variable property effects were minimized.

The measurements were focused on the determination of fullydeveloped heat transfer results . Both the heated and unheated port ions of the tube were heavily instrumented, which enabled bothcircumferentially local and average heat transfer resulted to be obtained. An appraisal of the circumferential average results yielded

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a criterion for the onset of buoyancy effects. Th e buoyancy-unaffectedaverage Nusselt numbers have been correlated as a function of theReyno lds and P rand t l numbers and a re , in add i t ion , compared toanalytical predictions for nonuniform heating and to empirical correlations for uniform heating. Circumferential d is tr ibutions arepresented for the Nusselt number, wall temperature, and heat f luxon the heated portion of the tube an d for the wall tem peratu re on theunheated portion of the tube. Comparisons are made between themeasured circumferential variations of the Nusselt num ber and hea tflux and those predicted by analysis .

A review of the literature revealed that only a few experiments havebeen performed on turbulent tube f low with circumferentially nonuniform thermal boun dary conditions [1-3] . ' Those experiments weregenerally of a more l imited scope than the present in that neitherbuoyancy effects nor the influence of heating orientation were investigated, the Prandtl number was restr ic ted to a s ingle value, andlow Reyno lds nu mbers were no t s tud ied . Fu r the rm ore , the ex ten t o fthe nonu niformity in heat f lux was not as great as that of the presentexperim ents . He at transfer coeffic ients were not reported in [2] , andthe length of the heated portion of the tube was only three diam eters .In [3] , sparse in stru me ntati on led to some uncertainti es in the resul ts .

The effect of c ircumferentially non uniform the rma l bounda ryconditions on turbulent forced convection heat transfer in tubes hasbeen analyzed in [4-7] for various turbulence models . The presentexperimental results will be compared with the predictions from themost recen t and en compassin g model [7] , which takes account ofdifferences in the tangential and radial eddy diffusivit ies near thewall.

Buoyancy effects on turbulent forced convection heat transfer inhorizontal tubes have, apparently , not been previously investigatedin the presence of c ircumferentially nonuniform therm al cond itions.For circumferentially uniform conditions, available flow regime maps[8-10] indicate that there are combinations of the Reynolds andRayleigh numbe rs where buoyancy effects are s ignificant.T h e E x p e r i m e n t s

The experimental apparatus is an open-loop flow circuit. The flowis delivered to the system by an elevated constant head tank whichis, in turn, fed by both hot and cold water l ines whose adjustmentenables control of the tank temperature (and, thereby, the Prandtlnumber). From the tank, the water passes downward into an insulatedinlet chamber which serves as a vertical/horizontal transit ion whileavoiding the secondary flows associated with an elbow or a bend. Thewater then is ducted throug h an 80-diamete r long hydro dynam icdevelopment section and into the electr ically heated test section.

! Numbers in brackets designate References at end of paper.

From the te st section, it passes through an exit section which is slightlyelevated to ensure that the test section and hydrodynamic development length are always filled with water. The discharge from the exitsection is directed into a weigh tank situated on a balance therebyenabling direct metering of the f low. The system is equipped withcut-off, by-pass, and control valves.

The test section, the hydrodynamic development length, and theexit section are all heavily insulated. In particular , the test sectioninsulation consists of a core of free-pouring silica aerogel powder(which envelops the tube) contained within walls made of polyure-thane sheet. Both the aerogel and the urethane have thermal conductivit ies less than th at of a ir . Heat losses were further combatedby the use of conical- t ipped p lastic supports for the test section.

The heart of the apparatus is the test section tube, which wasespecially designed and fabricated so that the fluid could be uniformlyheated over half of i ts c ircumference an d be unhea ted over the otherhalf. Since this heating condition has, apparently, not been previouslyinvestigated in the research l i terature , there were no precedents tofollow in designing the test section for the present experiments .Therefore, a substantia l amount of development work had to be undertaken and new fabrication procedures devised. These proceduresare descr ibed in de tail in Appe ndix A of [11], and only a brief description will be given here.

In s implest terms, the test section was fabricated by firs t takinglongitudinal cuts a long a thin-walled stainless s teel tube, therebydividing i t in to two portions. Th e circumference of one portion su btended an arc of 180 deg, whereas the circumference of the otherportion s ubtend ed a 150 deg arc (30 deg of circumference was removedby the machining). Then, the tube was re-formed about a circular rod,with a special epoxy being used to fill the two 15 deg gaps between therespective edges of the two portions of the tube. The re-formed tubehad an internal diame ter of 3 .07 cm (1.21 in .) and was 30 dia metersin length.

The tube had been internally honed prior to the longitudinal cuttingoperation. After re-forming, it was lightly honed again to remove anyepoxy tha t was not f lush with the s ta inless s teel wall . Th e end resultwas such that no discontinuity could be sensed when a finger, insertedthrough the end of the tub e, was moved over the epoxy joint .

The successful fabrication of the test section tube involved a myriadof factors encompassing design and fabrication of auxiliary parts ,numerous machining operations, ra tional selection of an epoxy andits proper curing, etc. Special attention had to be given to minimizingand com pensating for possible warpage and twisting of the tube dueto residual stress unlocked by the longitudinal cutting operation. Also,during the re-forming operation, the epoxy had to be maintained ati ts curing tempera ture (~120C ), necessita ting the heating of the rodabout which the tube was re-formed.

Once the test section tube had been fabricated, the internal diameters of the hydrodynamic development length and the exit sectionwere machined to conform to that of the re-formed tube. In this way,

N o m e n c l a t u r e .A = convective area at heated surfacecp = specific heat a t constant pressureD = inside diameter of tube, 2r ,g = acceleration of gravityk = thermal conductivity of water a t Ti nkw = thermal conductivity of tube wallm = mass flow rateNu = circumferential local Nusselt number,

equation (4)Nu = circumferential average Nusselt num

ber, equation (5)P r = P rand t l number a t T / )2Q = rate of heat transfer over 80 < 0 2)P rRe = Reynolds number, 2m/Virryr = radial coordinater , = inside radiusr , = mean radius, (r ; + r)/2r = outside radiusS = volume heat source due to ohmic heat

ingT = t e m p e r a t u r e

Ti,z = loca l bu lk tem pera tu reTi = inside wall temperature at 0Ti = average value of T; over 80 < 0 ^

80 T = ou ts ide wa l l t emper a tu re a t 0t = local wall th ickne sst = average thickness of heated wall2 = axial coordinateIi = thermal expansion coeffic ient a t '!),0 = angu la r coo rd ina teii = viscosity at Ti nv = kinematic viscosity at Tj,2

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continuity of the f low cross section was assured. Th e start in g lengthand the exit section were fabricated from pvc piping.

The heating of the test section was accomplished by passing a-celectr ic current longitudinally through the tube wall . The currentflowed in only one of the longitudinal halves of the test section, specifically, th at whose circum ference enco mpa sses 180 deg of arc. T h i swas achieved because the current carrying leads were attached onlyto tha t half and because the nonconduc ting epoxy preven ted t hecurrent from entering the other half.

In view of the low resistance of the test section (


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heated arc , i.e ., -9 0 deg < 0 < 90 deg, with T; being the circumferentia l average inner wall tem pera ture over tha t same arc and A = ;.However, as mentioned earlier , the temperature measurem ents in theneighborhood of 0 = 90 deg were of uncertain accuracy, and theiruse would have given rise to an uncer tainty in both Q and T;. Th erefore, to assure reliable resu lts , Q, T;, an d A were evaluated over thea rc -80 deg < 6 < 80 deg and, correspondingly, the N u values correspond to that arc .

In addition to the Nusselt number, the other dimensionless groupsthat are to be employed in the presentation of the results are theReyno lds , P ran d t l , and Ray le igh num bers

R e = 2m/nirn, P r = cpn/k, Ra = (g/35r;4/7e 2)Pr (6)In these groups, as well as in the Nu sselt num ber, a l l f lu id propertiesare evaluated at the bulk temperature T(. I t may be noted that inthe Rayleigh number, the grouping contained in the parentheses isthe so-called modified Grashof number which involves the surfaceheat f lux. The quantity q is an average hea t f lux defined as (irrm tS )/(irri), where the numerator is the heat input to the f luid at a givenaxial s ta tion and the denominator is the surface area of the heatedarc , both per unit axial length. The use of a Rayleigh number basedon heat f lux is motiv ated by the fact tha t the hea t f lux is nearly uni form on the heated arc .R e s u l t s a n d D i s c u s s i o n

C i r c u m f e r e n t i a l a v e r a g e r e s u l t s ; b u o y a n c y e f f e c ts . T h epresentation that follows will deal successively with the circumferentia l average results and the circum ferential local results . With regard to the average results , the f irs t task is to identify th e o peratin gconditions at which significant buoyancy effects were encountered.In this connection, there is persuasive evidence that none of thetop-heated runs were affected by buoyancy. This evidence has twofacets . First , for those conditions where the l ikelihood of buoyancyis greatest ( i .e . , the lowest Prandtl number and the lowest Reynoldsnumbers) , runs at different Rayleigh numbers did not show anyvariat ion in Nu beyond typ ical dat a scatte r [11]. Second, as will soonbe demonstrated, the Nu values for the low Reynolds number runsfall d irectly on Nu , Re curves that are logical extensions of results a th ighe r Reyno lds num bers .

In contrast to the foregoing, there are operating conditions wherethe bottom-heated runs were affected by buoyancy. This is demonstrat ed in Fig. 1. In this f igure, the ratio of the Nu sselt nu mb ers fortop and bo t tom hea t ing (Nu ( and Nui, ) a t the same Reynolds andPrand t l numbers i s p lo t ted a s a func t ion o f the Ray le igh number .Inasmuch as the Nusselt numbers at z/D = 20 and 25 typically agreedto within one percent, they were averaged and then plotted in thef igure . T he Reyno lds num ber i s the cu rve pa ramete r . T he P rand t lnum ber is not shown explicit ly because mixed convection is believedto depe nd on R e and Ra, but n ot Pr. Since Nui is not affected bybuoyancy, the f igure shows that Nut is buoyancy affected at lowReynolds numbers, with the effect being enhanced with increasingRay le igh number .

A Reynolds-Rayleigh criterion for the onset of buoyancy effects forbo t tom hea t ing was deduced f romFig . 1 and from ac rossp lo t the reo f[11]. The criterion was based on Nu(, /Nu ( = 1.05. Fig. 2 contains a R e,Ra diag ram w hich is subdivi ded into mixed convection and forcedconvection portions. As is physically reasonable , the threshold Rayleigh number for the onset of buoyancy effects increases with theR e y n o l d s n u m b e r .

At tem pts to compa re the results of Fig. 2 with l i teratu re information for c ircumferentially uniform thermal conditions did not provefruitful. Ne ithe r [8] nor [9] provide mixed convection threshold criteria which overlap the Re, Ra range of the present experiments andare based on circumferential average heat transfer results . Thethreshold cri teria of [10] are based on ten percent deviations (ratherthan five percent) and are reported in terms of a Rayleigh numberwhich involves the wall- to-bulk temperature difference (rather thanthe wall heat f lux).

Attention may now be turned to the buoyancy-unaffected circumferential average Nusselt number results , which are plotted as

Fig. 1 Ratio of circum ferential average Nu sselt numb ers for bottom and topheating at low Reynolds numbers

4

2

I0 786

4

2

in 6

- 7.67.47.2

- 7.0o-


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600

400

200

10080

Nu 6040

20

o TOP HEATINGh a BOTTOM HEATING

Pr = 3. 6

4 0 6 0 8 0Re xlO'

Fig. 3 Circum ferent ial average Nusselt numbers for buoyancy-u naffectedturbulent flow

N uN u n o

R e x l O 'Fig. 4 Co mparison of present Nu results with literature Information

for c ircumferentially nonuniform boundary conditions was employedby the prese nt auth ors to compute N u for the range of 6 between 80and 80, which is the sam e range for which the expe rimenta l Nu valueswere evaluated.

The comp arisons are show n in Fig. 4 . To m ute the variation of theresults with Reynolds number and to achieve ordinate values on theorder of one, the Nusselt numbers have been normalized by thosecomputed from the Dittus-Boelter equation. Since the l i terature information is not in tend ed to be applied for Re below 10,000, the curveshave been te rmin a ted a t th a t po in t .

From an overall appraisal of the f igure, i t is seen that none of thepredictions from the l i terature , including those based on circumferentia lly uniform cond itions, are very far from the pres ent data . T hisis s t i l l anothe r affirmation of the forgiving nat ure of turbul ent f low.The general level of agreement is on the order of ten percent. Thepredictions of [7] , which take acc ount of c ircumferential nonunifor-mities , l ie paralle l to the present data and are displaced below themby about f ive percent. I t is a lso interesting to note the general consensus in evidence in the figure that the Dittus-Boelter equation givesNusselt numbers that are too low at the higher Reynolds number.

C i r c u m f e r e n t i a l l o c a l re s u l t s . T he re su l t s tha t wil l now bepresented include circumferential d is tr ibutions of the local Nusseltnumber, temperature, and heat f lux on the heated arc and the distr ibution of tempe ratu re on the adiab atic wall . A comparison of thesequantit ies with th ose com puted using the prediction procedure of [7]will also be presented. Owing to space limitations, only representativeresults will be given here, but a comp lete prese ntation is available in[11].Local Nusselt number results for both top heating and bottomheating are plotted in Fig. 5 as a function of angular position between-8 0 a nd 80 . For top heating, 0 = 0 corresponds to the to pmo stposit ion on the circumference, whereas for bottom heating 6 = 0 isthe bottom -mo st posit ion. The figure is subdivide d into three parts ,respectively, for Pr = 3.6, 5.7, and 9.4. Each part contains a stack ofgraphs corresponding to successively increasing values of Reyno ldsnumber. D ata p oints are shown only in a few graphs for i l lustration ,but are otherwise omitted to preserve clari ty .

Attention will be focused on the results for Pr = 3.6, which portraya sequence of events that is a lso in evidence for the other Prandtlnumb ers, but to a lesser extent. At the lowest Reynolds n um ber, the

P r = 3 . 6 Pr = 9.4

1.0

1.0

1.0

1.0

NuNu2 . 0

1.0

R e = 4 3 , 0 0 0

2 0 , 6 0 0

10,300

5 2 4 0

- o TOP HEATING- - 0 - - BOTTOM HEATING

Re = 10,6005 0 7 03 0 5 0

- 1.0

1.0

1.0

- 8 0 - 4 0 0 4 0 8 0 - 8 0 - 4 0 0 4 0 8 09 ( D E G R E E S )

Fig. 5 Circu mferential distribu tions of the local Nu sselt numb er

bottom -heated Nusselt numb er distr ibution is nearly uniform owingto the circumferential transport provided by the buoyancy-inducedsecondary f low. On the other hand, the distr ibution for top heating

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is d is t inctly nonuniform 3 inasmuch as buoyancy is absent and thecircumferential turbulen t transpo rt is weak. As the R eynolds num berincreases, the buoyancy is rende red ineffectual and th e circum ferentia l turbulent transport grows stronger. These factors cause thetop-heating and bottom-heating results to come together andmerge.

Represen ta t ive d is t r ibu t ions o f in s ide wa ll t empera tu re and hea tflux on the hea ted arc are presented in Pig. 6 , respectively for Pr =3.6 ( left-hand portion) and Pr = 9 .4 (r ight-hand portion). For eachPrandtl number, results are shown for Re ~5,000 and 40,000. The walltemperature and heat f lux distr ibutions are respectively expressedas (T; Tbz)l(Ti - Tbz) and q/q, where both T; and q are functionsof 6, and T; and q are aver ages over 80 deg < 0 < 80 deg. Th e d atafor top heating are portrayed by circles , whereas those for bottomheating are portrayed by squares. The actual data points are plottedfor all cases in preference to showing faired curves since, in manycases, the faired curve would be a horizontal line at an ordinate of1.0.

Inspection of Fig. 6 shows tha t the he at f lux is very nearly circum -ferentially uniform for all of the cases. On the other hand, the walltemperatures exhibit various degrees of nonuniformity. At Pr = 3 .6and Re ~ 5,000, the top-heate d (buoyancy-unaffected) distr ib utionis dist inctly nonuniform, w hereas the bottom-h eated (buoyancy-affected) distr ibution is uniform. With increasing Reynolds num ber,the dist inction between top and bottom heating disappears , and theshape of the distribution for Re ~ 40,000 is typical of that for all ofthe buoyancy-unaffected, fully turbulent cases.

For Pr = 9 .4 and Re ~ 5,000, the distr ib utions for top heating andbottom heating are coincident, thereby reflecting the absence ofbuoyancy effects . At Re ~ 40,000, the tem per atur e distr ib ution po ssesses the aforementioned universal shape.

Circumferential d is tr ibutions of the temperature on the unheatedwall are plotted in Fig. 7 for the same cases and w ith the same forma tas was used in Fig. 5 for the N usselt nu mbe rs on the heat ed arc . Th eangular coordinate 8 = 180 deg marks the circumferential m id-point

of the unhe ated wall and is a lso the mid- point of the abscissa of Fig7. The ord inate compare s the wall- to-bulk tempe ratur e differenceat a point on the unheated wall with the average wall- to-bulk temperat ure difference on the hea ted arc. A negative value of the o rdinateindicates tha t the local wall tem pera ture is lower tha n the bulk temperature, and vice-versa. As before, the results for Pr = 3.6 are themost de mon strative and are , therefore, s ingled out for discussion.

If a t tention is turned to the lowest Reynolds number (~3,000), i tis seen that for top heating (solid line) the adiabatic wall temperatureis lower than the bulk, whereas the opposite relation is in evidencefor bottom heating (dashed l ine). This is another c lear affirmationof the presence of buoyancy for the bottom heating case and of itsabsence for top he ating. Th e buoyan cy, when pres ent, g ives r ise to asecondary flow which carries hot fluid into the upper portion of thetube. Th e shape of the dash ed curve suggests a secondary flow patternwhereby hot fluid moves upward along the walls of the two verticalhalves of the tube a nd de scends along the vertical diam eter. As theReynolds number increases, the buoyancy-induced secondary f lowdiminishes. The distr ibution curves for top and bottom heating drawtogether and are characterized by adiabatic wall temperatures thatare lower than the bulk.

As a final matter, attention will be turned to the application of theprocedu re of [7] for predicting the circum ferential variations in theabsence of buoyancy. To use that method, the inside wall temperaturedistributions (over the entire 360 deg arc) were fit with a Fourier ser ies . Th e Fourier coefficients were then em ployed, a long with tabulated influence co

j.heat.transfer.1978.vol.100.n3

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