International Communications in Heat and Mass Transfer 54 (2014) 27–32

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International Communications in Heat and Mass Transfer

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Diameter effect on supercritical heat transfer☆

S. Yildiz ⁎, D.C. GroeneveldDepartment of Mechanical Engineering, University of Ottawa, Ottawa, ON K1N 6N5, Canada

☆ Communicated by W.J. Minkowycz⁎ Corresponding author at: Department of Mechanica

University, 34349 Yildiz, Istanbul, Turkey.E-mail address: syildiz@yildiz.edu.tr (S. Yildiz).

http://dx.doi.org/10.1016/j.icheatmasstransfer.2014.02.010735-1933/© 2014 Elsevier Ltd. All rights reserved.

a b s t r a c t

a r t i c l e i n f o

Available online 6 March 2014

Keywords:Diameter effectSuper critical heat transferDeterioration

The objective of this review is to assess and analyze the literature on the effect of tube diameter on heat transfer atsuper-critical (SC) pressures. The review is based on SC heat transfer data obtained in tubes with a diameterrange of 3.18 to 38.1 mm, cooled by carbon dioxide, water, R-22, and R-12. The majority of experimental studiesshow that, for the same flow conditions, the heat transfer coefficient (HTC) in the ‘normal’ heat transfer modeincreases with a decrease in tube diameter. Furthermore, it was found that at SC pressures, heat transfer ismore prone to deteriorate in large tube diameters. In the “deteriorated” heat transfermode, the HTC also appearsto decrease with an increase in tube diameter.

© 2014 Elsevier Ltd. All rights reserved.

1. Introduction

The objective of this review is to improve our understanding of thediameter effect on supercritical heat transfer (SCHT). The diameter ef-fect will be examined for each of the three types of heat transfermodes at SC pressures, i.e. (i) normal heat transfer, which is similar tothat of subcritical region away from the critical or pseudo-critical re-gions, (ii) enhanced heat transfer which provides more efficient heattransfer compared to that in the normal heat transfer region, and(iii) deteriorated heat transfer, which provides less efficient heat trans-fer compared to that in the normal heat transfer region (Pioro andDuffey [1]). In the ‘normal’ heat transfer mode, the heat transfer canbe predicted using conventional single-phase Dittus–Boelter typecorrelations.

Ackerman [2] and others postulate a boiling-like process occurringat SC pressures (similar to sub-cooled nucleate boiling at subcriticalpressure) that gives rise to the sudden increase in the HTC over that ofnormal forced convection. Lee and Haller [3] also observed improvedheat transfer, especially at temperatures very close to pseudo critical.These conditions occur at high mass flows. When the bulk fluid isclose to the pseudo critical temperature, agglomerations of vapor-likefluid cells detach from the wall and break up the laminar boundarylayer, thus resulting inmore turbulence and consequently a higher HTC.

Heat transfer deterioration (HTD) is characterized by a sudden de-crease in the heat transfer coefficient (HTC) or a sharp increase in thewall temperature. Pioro and Duffey [1] cite references discussing twoplausible types of HTD occurring at supercritical pressures. The first

l Engineering, Yildiz Technical

7

type occurred at the entrance section of a tubular test section at lowmass fluxes and high heat fluxes, whereas the second type occurred inthe tube at locations where the wall temperature exceeded thepseudo-critical value. The mechanism for the first type of HTD is stillnot well understood. Explanations for the second type of deteriorationinclude the following two: (i) the occurrence of “pseudo-film boiling”(similar to film boiling at subcritical pressures), where a low-densityfluid layer forms near the wall and prevents the high-density bulkfluid from rewetting the heated surface (Ackerman [2]), and (ii) strongvariation in thermo-physical properties near the pseudo-critical tem-perature (Wang et al. [4]).

Wang et al. [4] reported that heat transfer deterioration should beavoided in power plants operating at SC pressures, whether these be nu-clear reactors or fossil-fuelled plants. The consequences of HTD (oroverheating of the heated surface) could include accelerated corrosionand/or failure of the heated surface. Wang et al. [4] compared the heattransfer characteristics of SC pressurewater to that of subcritical pressurewater in vertically-upward flow in tubes. They also noted that the HTD ofSC water is similar in mechanism to post-CHF heat transfer at subcriticalpressures: both are caused by the blanketing of the heated surface by alow-density fluid. This blanket will have a lower thermal conductivitythen the bulk of the fluid and will cause the wall temperature to rise.

In the case of subcritical pressures, the CHF and film boiling heattransfer are primarily affected by themass flux, pressure, fluid enthalpy(or inlet subcooling and heated length), and tube diameter. Accordingto Collier and Thome [5], at constant local thermodynamic quality, theCHF decreases with increasing tube diameter. In case of SC pressures,Lee and Haller [3] found that heat flux and tube diameter were the im-portant parameters that affect the lower mass-flux limits below whichpseudo-film boiling will occur. Song et al. [6] noted that the flow inthe larger diameter tube was prone to deterioration in the heat transferdue to buoyancy as discussed in Section 5.

Nomenclature

Cp specific heatd diameterg acceleration due to gravityG mass fluxh heat transfer coefficientH enthalpyk thermal conductivityL lengthNq heat loading factorP pressureq heat fluxT temperature

Greekρ densityμ viscosity

Dimensionless numbersBu BuoyancyEc Eckert numberGr Grashof numberNu Nusselt numberPr Prandtl numberRe Reynolds number

Subscriptsb at bulk fluid temperatureeq equivalentexp experimentalF at film temperaturepc at pseudo-critical temperaturevarp at variable physical propertiesw at wall temperature

AbbreviationsDNB departure from nucleate boilingHT heat transferHTC heat transfer coefficientHTD heat transfer deteriorationID inside diameterOD outer diameterODHT onset of deteriorated heat transferSC super critical

Fig. 1. Diameter effect on the HTC in ‘normal’ heat transfer region according toLoewenberg [13] look-up table for water.

28 S. Yildiz, D.C. Groeneveld / International Communications in Heat and Mass Transfer 54 (2014) 27–32

The post-CHF heat transfer at sub-critical pressures and the deterio-rated heat transfer at SC pressures are critical parameters in the safetyanalysis of a nuclear reactor. It is therefore important to determine theeffect of diameter on SCHT.

In this review we will first review the papers on diameter effect onnormal and deteriorated heat transfer at SC pressures, then discustheir findings and the similarity between sub- and supercritical heattransfers, and finally provide some overall conclusions.

2. Diameter effect in normal heat transfer

Song et al. [6] investigated the SCHT of a vertical upward flow of car-bon dioxide in two different diameter tubes; 4.4 and 9.0 mm ID. At the

high mass flux, only normal heat transfer was encountered. They notedthat the HTC in the larger ID tube is consistently lower than that in thesmaller ID at bulk temperatures below and above the pseudo-criticaltemperature.

Kim et al. [7] used the same equipment and experimental conditionsof Song et al. [6], and found the same results. In addition, they tested aconcentric annular flow geometry having an 8 mm ID and 10 mm OD.For the annular test section, the equivalent diameter (based on the heat-ed perimeter) was almost identical to the diameter of the smaller tube(4.4 mm ID). The HTC for the annulus was slightly lower than that forthe 4.4 mm ID tube.

Watts and Chou [8] examined mixed convection heat transfer towater at SC pressure using two different diameter tubes, 25.4 mm IDand 32.2mm ID respectively. They proposed correlations formixed con-vection (see Table A1) that include the diameter effect. The tendency ofthese correlations is to decrease HTC with an increase in diameter.

Yamashita et al. [9] investigated SCHT for upflow in a uniformly heat-ed vertical 4.4 mm ID tube using R-22 as working fluid. The resultswere compared with previous data obtained in larger diameter tubes(Yoshida [10], Yoshida et al. [11]). For ‘normal’ heat transfer, the HTCwas larger with the smaller diameter tube, i.e. the HTC trend with thetube diameterwas similar to that of theDittus–Boelter type correlations.

Mayinger and Scheidt [12] investigated SCHT in R-12 cooled tubes(12.5 and 24.3 mm ID) with vertical upflow. For the “normal” heattransfer' regions, no effect of diameter on HTC was observed for thesame flow conditions. They proposed correlations for predicting theheat transfer under conditions in the “normal” region (Table A1).

Loewenberg [13] introduced a look-up table to predict heat transferof supercritical water. According to the look-up table, HTC decreaseswith an increase in diameter in the ‘normal’ heat transfer, as shownby the Dittus–Boelter correlation in Fig. 1.

3. Diameter effect on Onset of Deteriorated Heat Transfer

The Onset of Deteriorated Heat Transfer (ODHT) is strongly affectedby both mass flux and heat flux, as is often illustrated graphically (e.g.Ackerman [2], Lee and Haller [3], Yamashita et al. [9], Loewenberg[13]); the ODHT can also be predicted using criteria for the buoyancy pa-rameter (Bae et al. [14], Jackson andHaller [15]) aswill be shown below.

Ackerman [2] showed the ODHT boundary on a mass velocity vs.heat flux plots similar to Fig. 2. Note that DHT (or pseudo-film boiling)occurs in the region below and to the right of the line for a given diam-eter, while the upper left region corresponds to normal forced-convection heat transfer. An increase in diameter will move the ODHTboundary up towards the left-hand corner, i.e. DHT will then occur ear-lier at a lower heat flux and/or mass flux. Ackerman [2] reported thatthe heat flux at ODHT increased 40%with the 9.4 mm ID tube comparedto the 24.4 mm ID tube. For a given mass velocity, an increase in pres-sure above the critical pressure allows operation at higher heat fluxes

Fig. 2. Effect of tube diameter on the limit heat flux.

29S. Yildiz, D.C. Groeneveld / International Communications in Heat and Mass Transfer 54 (2014) 27–32

without pseudo-film boiling; in general, as pressure increases the limitcurve shifts to the right.

Lee andHaller [3] proposedmass velocity limits for ODHT for a givenheat flux and tube diameter for fluid temperature below the pseudo-critical temperature. Their graph was similar to that proposed byAckerman [2]. For a given heat flux, larger diameter tubes requiredhigher mass flows to avoid the deterioration in heat transfer.Loewenberg [13] schematically showed a relationship between massflux and limit heat flux with the tube diameter as a parameter, whichwas similar to Fig. 2.

Yamashita et al. [9] graphically showed the relation between thelimit heat flux at ODHT and bulk fluid enthalpy with the tube diameteras a parameter. The limit heat flux is larger for the smaller diametertube. However, near the pseudo-critical point, the difference in thelimit heat flux is small. With increasing bulk fluid enthalpy, the limitheat flux generally decreases.

Song et al. [6] reported that their results are consistent with thetrends shown in Fig. 2. In case of lowmass flux and low heat flux, no de-terioration occurred for the 4.4 mm ID while for the 9.0 mm ID showedthat some HTD occurred. Fig. 3 compares the HTC from Song et al. [6]with the Dittus–Boelter correlation and the correlation proposed byWatts and Chou [8] and shows similar trends in the ‘normal’ heat trans-fer region. In deteriorated heat transfer region,Watts and Chou correla-tion shows a good agreement with the experimental HTC.

Kim et al. [7] observed that at lowmass fluxes in a 9.0 mm tube, de-teriorated heat transfer occurred while the deterioration was sup-pressed in the annulus and in the 4.4 mm tube for the same flowconditions. According to Kim et al. [7], heat transfer in the large IDtube is more susceptible to deterioration than the flow in the small IDtube. The effect of the increased heat loading factor (Nq = qdeq/kbTb)outweighed the increase in heat transfer due to the increased Reynoldsnumber in the large diameter tube.

Bae et al. [14] examined forced andmixed convection heat transfer tosupercritical carbon dioxide during up and downflow in a 6.32 mm ID

Fig. 3. Diameter effect on the ODHT according to data from Song et al. [6].

tube. They compared their results with previous Korean data for 4.4and 9.0 mm ID tubes. For the low heat flux, HTD developed in the9.0 mm ID tube while normal heat transfer remained in the 4.4 mmand 6.32 mm ID tube. Bae et al. [14] concluded that the ODHT startswhen the buoyancy parameter Bu N 2.0 × 10−5 where Bu ¼ Grb=Reb

2:7.According to Jackson et al. [16], using the buoyancy parameter, data

on severe deterioration in local heat transfer can be used to estimate theReynolds numbers at which similar effects may be expected with tubesof reduced diameter. According to Jackson et al. [16], the impairment ofheat transfer is related to buoyancy effects and is caused by partiallaminarization of flow due to reduction of shear stress across the buoy-ant layer adjacent to the heated wall, reduced turbulence and conse-quent impaired turbulent heat diffusion. Jackson and Haller [15]determined that the HTD occurs at Bu N 10−5. Table 1 summarizes theproposed criteria for ODHT which includes a tube diameter effect.

4. Diameter effect on deteriorated heat transfer

Ackerman [2] investigated the SCHT to water at supercritical pres-sures in smooth vertical tubes (9.4, 11.94, 18.5, and 24.4 mm ID) and aninternally ribbed tubes (18.0 mm ID). Ackerman noted that anomaloustrends in heat transfer were sometimes observed when Tb b Tpc b Tw: asudden decrease in HTC below that for normal forced convection. Thiswas attributed to a deteriorated heat transfer (DHT) and was similar tofilm boiling at subcritical pressure. At high inlet fluid temperature forTw N Tpc, the wall temperature increased disproportionally with heatflux for the 18.5 mm ID tube. As the heat flux was increased further, apeak in wall temperature becomes more pronounced and moved up-stream. A similar trend was observed for the 24.4 mm ID tube.

Shiralkar and Griffith [17] studied HTD of SC carbon dioxide in twovertical test sections having a 3.18 and 6.35 mm ID. The wall tempera-ture peaks for the 3.18 mm tube were not as sharp as those for the6.35 mm tube (the smaller diameter reduced the deterioration in heattransfer).

Watts and Chou [8] proposed correlations for deteriorated heattransfer (Table A1): according to their correlations, the HTC decreaseswith an increase in diameter. Bae et al. [14], however, determinedthat, in theDHT region, theHTC for the large diameter tubeswas slightlyhigher than for the small diameter tubes up to a certain bulk fluid en-thalpy. Mayinger and Scheidt [12] also found that in the DHT regionwith large diameter tubes, the wall temperature increase was slightlyhigher than for the smaller diameter tubes. However, Yamashita et al.[9] noted that in the DHT region, no effect of tube diameter on heattransfer was observed.

5. Discussion

Table 2 shows parameter ranges of the experiments investigating thediameter effect on SCHT. Although the overall experimental parameterranges can be quite wide, the authors investigated the effect of diameter

Table 1Proposed ODHT Criteria.

Author ODHT criteria

Bae et al. [14] Bu ¼ Grb=Reb2:7≥2:0� 10−5

Ackerman [2] ODHT limit curve was not quantified; onlya sketch was provided.

Lee and Haller [3] Graph for ODHT mass velocity limits.Loewenberg [13] Not quantified; provided only a sketch for

the relation between the ODHT limit heatflux and mass flux with tube diameteras a parameter.

Yamashita et al. [9] Graph for ODHT limit heat flux and thebulk fluid enthalpy with the tube diameteras a parameter.

Jackson and Hall [15] Bu ¼ Grb=Reb2:7 N10−5

30 S. Yildiz, D.C. Groeneveld / International Communications in Heat and Mass Transfer 54 (2014) 27–32

on SCHT over amuch narrower range, e.g. the pressure ranges for each ofthe diameter effect studies were very narrow,making it difficult tomakedefinite conclusions on the diameter effect for different pressures.

Most of the authors noted an increase in HTCwith a decrease in tubediameter in the normal heat transfer mode. This was expected since theHTC is proportional to di

−0.2 according to the Dittus–Boelter type corre-lations for single-phase heat transfer.

As shown by Pioro and Duffey [1] and others, HTD usually appears athigh heat fluxes and lowmass fluxes. Most of the authors found that anincrease in diameter lowers the heat flux required to initiate deteriora-tion in heat transfer. According to some authors, the buoyancy parame-ter is the main criteria for ODHT (Jackson et al. [16], Bae et al. [14]). Thebuoyancy parameter is defined as:

Bu ¼ Grb=Reb2:7 ð1Þ

where

Grb ¼ ρb ρb−ρð Þgd3μb

2 ð2Þ

Reb ¼ Gd=μb ð3Þ

From Eq. (1) to (3), Bu ~ d0.3 hence the ODHT will be delayed insmaller diameter tubes. This can be explained by considering the pres-ence of both free and forced convection, i.e. mixed convection. Increas-ing buoyancy increases the level of free convection. For the same flowconditions, the buoyancy effect is stronger in a larger diameter tubes,and can lead to a deterioration of the heat transfer. According to Baeet al. [14], the HTD commences when the buoyancy parameter reached2.0 × 10−5.

The acceleration effect is due to an expansion of the fluid, which, forthe same G, q and axial length would be much greater in a small diam-eter tube. Thiswould lead tomore turbulence and better heat transfer insmaller diameter tubes. In addition, large changes in density betweenthe wall and bulk, as they occur in the case of Tb b Tpc b Tw can generatea strong buoyancy effects in larger diameter tubes. This could result inan M-shaped velocity profile, but will also increase the thickness ofthe low density layer near the wall in larger diameter tubes, thus

Table 2Details of experiments investigating the diameter effect on SCHT.

Author Heated lengthm

Test sectionID or Deq,mm

Conditions

Mass fluxkg/m2s

Song et al. [6] 2.12.65

Tube A: 4.4Tube B: 9.0

Case 1Case 2Case 3

Kim et al. [7] 2.12 Tube A: 4.4 Same as Son2.65 Tube B: 9.01.8 Annulus E: 4.5

Bae et al. [14] 2.65 Tube, 6.32 285–1200

Shiralkar and Griffith [17] 1.5 6.35 and 3.18 Re2.67 × 105

8.35 × 1051.5 Swirl, 6.35

Jackson et al. [16] 2.45 19 Re2 × 105–2 ×

Ackerman [2] 1.83 9.4, 11.94, and 24.4 136–21702.74 18.51.83 Ribbed, 18.0 407

Lee and Haller [3] 4.57 38.1 and 37.7 542–2441Watts and Chou [8] 2 25.4 and 32.2 132–1060Yamashita et al. [9] 2 4.4, 9.0, and 13.0 400–2000Mayinger and Scheidt [12] 6 12.5 and 24.3 375–1255

a The critical pressure and temperature for water are 22.06 MPa and 373.95 °C; CO2: Pcr =Tcr = 112 °C

causing an additional deterioration in the heat transfer. The above qual-itative explanation is consistent with the observed reduction in heattransfer in larger diameter tubes.

6. Similarity in diameter effect for subcritical and supercriticalheat transfer

Since the heat transfermechanisms in the near critical region at sub-critical and SC pressures are similar, it was thought useful to comparethe diameter effect on heat transfer for sub- and super-critical pres-sures. Table 3 summarizes the observed similarities in diameter effectbetween the SCHT and the subcritical HT.

6.1. Normal heat transfer

The heat transfer coefficient at subcritical pressure decreases withan increase in diameter for single-phase flow. This is identical withthe trend of HTC at SC pressure. In both cases the HTC is roughly propor-tional to di

−0.2 according to Dittus–Boelter type correlations.

6.2. CHF and onset of deteriorated heat transfer

According to Tanase et al. [18] andWong [19], the heat flux requiredto initiate heat transfer deterioration at subcritical pressures (i.e. CHF) de-creases with increasing tube diameter at the constant local conditions (P,G, X or enthalpy) at subcritical pressures in two phase flow. This agreeswith Bae et al. [14] who reported that, for a given heat flux, mass flux,pressure, and local enthalpy, the ODHT at SC pressures occurred firstwith a larger diameter tube whereas the smaller diameter tube requireda higher heat flux to initiate HTD. In addition, Ackerman [2], Lee andHaller [3], Yamashita et al. [9], and Loewenberg [13] all reported an in-crease in ODHT limit heat flux with decreasing tube diameter.

For the same pressure,mass flux and subcooling, with smaller diam-eter tubes radial velocity gradients become larger. Larger velocity gradi-ent lead to an increase in the rate of detachment of the growing bubblesat subcritical pressures. As a result, the CHF increases. The larger radialvelocity gradients may help to move agglomerates of low density fluidfrom the near-wall region into the main stream, and delay ODHT withsmaller diameter tubes at SC pressures.

PMPa

P/Pca Workingfluid

Heat fluxkW/m2

Flowdirection

400 30 Up 8.12 1.1 CO2

400 501200 50g et al. (2008) Up 8.12 1.1 CO2

30–170 Up/down 7.75, 8.12 1.05,1.1

CO2

50–452 Up/down 7.58, 7.93 1.03,1.07

CO2

10459–0.75 Up/down – 1.12,0.88 CO2

126–1730 – 22.8–41.3 1.03–1.87 Water

315–631 – 24.82 1.13250–1570 Up 24.1 1.09 Water175–440 Up/down 25 1.13 Water10–170 Up 5.5 1.12 R-228.6–78 Up 4.28, 4.5 1.03,

1.09R-12

7.38 MPa, Tcr = 30.98 °C; R-22: Pcr = 4.9 MPa, Tcr = 95.19 °C; R-12: Pcr = 4.14 MPa,

Table 3Similarity in diameter effect on HT at subcritical and SC pressures.

HT Effect of increase in diameter

Subcritical pressure (P b Pc) SC pressure (P N Pc)

Normal • Decrease in HTC in the single-phase flow (HTC ~ ID−0.2

as predicted by Dittus–Boelter type correlation)• Decrease in HTCHTC ~ ID−0.2

(Dittus–Boelter type correlation)CHF/ODHT limit heat flux Decrease in CHF (Tanase et al. [18], Wong [19],

Collier and Thome [5])• Decrease in the heat flux limit at ODHT (Song et al. [6], Kim et al. [7], Bae et al. [14],Ackerman [2], Lee and Haller [3], Yamashita et al. [9])

Deteriorated/film boiling/pseudo-film-boiling

• Decrease in HTC (HTC ~ ID−0.2) as shown by thefilm boiling look-up table (Groeneveld et al. [20])

• Decrease in HTC (Watts and Chou [8], Shiralkar and Griffith [17])• Increase in HTC (Bae et al. [14], Mayinger and Scheidt [12])• No effect in HTC (Yamashita et al. [9]).

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6.3. Film boiling and deteriorated heat transfer

Several authors have commented on the similarity of film boiling atsubcritical pressure (where a sudden decrease in HTC occurs when theCHF is exceeded), and pseudo-film boiling or DHT at supercritical pres-sures (where a similar but less severe decreases in HTC are observed).Pressure, bulk fluid enthalpy, mass flux, heat flux and tube diameter allaffect the film boiling and the pseudo-film boiling HTC (Ackerman [2]).

According to the correlationsproposed byWatts and Chou [8] for de-teriorated heat transfer, HTC decreases with an increase in diameter atSC pressure. This agreeswith the film boiling look-up table at subcriticalpressures (Groeneveld et al. [20]).

On the other hand, HTC increases with an increase in diameter (Baeet al. [14], Mayinger and Scheidt [12]). Furthermore, no diameter effecton the HTC was observed by Yamashita et al. [9].

The impact of the diameter on the deteriorated heat transfer de-pends also on the magnitude of the heat flux; at high and at low heat

Correlations for deteriorated heat transfer at supercritical pressures.

Author Correlations

Watts and Chou [8] Nu varp ¼ 0:021Reb0:8 Prb

0:55ρw=ρbð Þ0:35

For normal heat transfer:

Nu = Nuvar p for Grb=Reb2:7 Prb

0:5b10−5

Nu ¼ Nuvarp 1−3000Grb=Reb2:7 Prb

0:5h i0:295

for 10−5

Nu ¼ Nuvarp 7000Grb=Reb2:7 Prb

0:5h i0:295

for

Grb=Reb2:7 Prb

0:5N10−4

For deteriorated heat transfer:

Nu ¼ Nuvarp 1:27−19500Grb=Reb2:7 Prb

0:5h i0:7

for

Grb=Reb2:7 Prb

0:5b4:5� 10−5

Nu ¼ Nuvarp 2600Grb=Reb2:7 Prb

0:5h i0:305

for

Grb=Reb2:7 Prb

0:5N4:5� 10−5

Grb ¼ ρb ρb−ρð Þgd3μb

2 ; Reb = G.d/μb; Prb ¼ μbCp=kb ; Cp ¼Mayinger and Scheidt [12] Ec N 1; Tb b Tw b Tpc

NuF = C1ReF0.85 Pr F0.33(ρw/ρb)0.20

1 ≥ Ec ≥ 0; Tb ≤ Tpc ≤ Tw

Nub ¼ C2Re0:87b Pr0:56b ρw=ρbð Þ0:21 Cp=Cpb

� �0:57Ra0:06b

Ec b 0; Tpc b Tb b TwNuF = C3ReF0.85 Pr F

0.62(ρw/ρb)0.155

Under deteriorated conditions:

Nub ¼ C4Re0:87b Pr0:61b ρw=ρbð Þ0:18 Cp=Cpb

� �0:28Ra0:12b

Ec = (Tpc − Tb)/(Tw − Tb); Cp ¼ 1Tw−Tb

∫Tw

TbCp T ; Pð Þd

Grb ¼ gβb Tw−Tbð Þd3ν2b

; βb ¼ ρbT2−T1

1ρ T2ð Þ− 1

ρ T1ð Þ� �

; Rab = G

R12 : C1 : 0:0102;C2 : 0:00166;C3 : 0:0094; C4 : 0:0Water : C1 : 0:0128;C2 : 0:00207;C3 : 0:0115;C4 : 0

fluxes the effect of diameter in the deteriorated heat transfer at super-critical pressures shows different trends.

7. Conclusions and final remarks

1. For “normal” heat transfer, theHTC is higher in small diameter tubes,and follows the trends predicted by Dittus–Boelter type correlations.

2. For subcritical pressures, the trend predicted by the film boiling look-up table shows a clear decrease in heat transfer with an increase indiameter. For SC pressures, the diameter effect on DHT is less clear.Although there appears to be some preference for a decrease inHTCwith an increase in diameter, some studies show either no effectof diameter on the deteriorated HTC or a small increase in HTC withan increase in diameter.

3. Various criteria have been proposed for predicting the ODHT, theygenerally show that theODHT occurs at a lower heatflux in larger di-ameter tubes. This is similar to the CHF at subcritical pressures.

Table A1

Appendix A

Parameter

bGrb=Reb2:7 Prb

0:5≤10−4

1Tw−Tb

∫Tw

TbCp T ; Pð ÞdT ¼ Hw−Hb

Tw−Tb; ρ ¼ 1

Tw−Tb∫Tw

TbρdT

Waterd = 25,4; 32.2 mmL = 2 mP = 25 MPaG = 106–1060 kg/m2sq = 175–440 kW/m2

Tb-inlet = 150–310 °CTw-inside = 260–520 °CUp/down

T ¼ hw−hbTw−Tb

rb Prb

0038:00024

R-12d = 12.5, 24.3 mmL = 6 mP = 25 MPaG = 375–1255 kg/m2sq = 8.6–78 kW/m2

32 S. Yildiz, D.C. Groeneveld / International Communications in Heat and Mass Transfer 54 (2014) 27–32

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