A correlation for heat transfer coefficient during stratified steam condensation in large flattened tubes with variable inclination and wall temperature

International Journal of Heat and Mass Transfer 146 (2020) 118666

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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

A correlation for heat transfer coefficient during stratified steam condensation in large flattened tubes with variable inclination and wall temperature William A. Davies III a, Pega Hrnjak a,b,⇑ a b

Air Conditioning and Refrigeration Center (ACRC), Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, IL, USA CTS – Creative Thermal Solutions, Inc., Urbana, IL, USA

a r t i c l e

i n f o

Article history: Received 6 May 2019 Received in revised form 18 July 2019 Accepted 31 August 2019

Keywords: Steam condenser Heat transfer coefficient Experimental facility Heat transfer Condensation Flattened tube

a b s t r a c t This paper experimentally investigates heat transfer coefficient during stratified-flow condensation in a large flattened-tube steam condenser with non-uniform heat flux and wall temperature, and varying inclination angle. The heat transfer test facility is designed and built to provide local measurements of wall temperature; combined with a CFD model, it also provides local heat flux and heat transfer coefficient. The condenser test tube is that commonly used in air-cooled condensers for power plants. The steel tube has inner dimensions of 216 mm height
16 mm width. In addition, tube inclination is varied from 0° to 38° downwards. The test facility is designed to match the conditions of an operating condenser, with low steam mass flux, realistic development of flow regimes and a large temperature glide on the cooling side. The results show that heat transfer coefficient along the tube wall follows the Nusselt condensation model, while heat transfer through the stratified liquid layer at the tube bottom is predominantly driven by laminar forced convection. Commonly-used correlations for heat transfer coefficient are unable to accurately predict the experimental results, so a new correlation is proposed. The new correlation takes into account heat transfer through the stratified liquid layer at the tube bottom as well as through the condensing film along the tube wall. Recommendations are made for situations where: (1) the wall temperature is known and (2) the wall temperature is unknown. Finally, a recommendation is made for combining the correlation in the stratified condensate layer with a local model for determining condenser capacity. Ó 2019 Elsevier Ltd. All rights reserved.

1. Introduction Large, flattened-tube steam condensers are commonly used for air-cooled power plants [1]. Their use has increased significantly in the 21st century [2], and there has been a likewise increase in research investigating the condenser design and governing physics. However, there are no experimentally-validated correlations for steam-side heat transfer coefficient (HTC) for flattened-tube aircooled condensers (ACCs), which causes significant uncertainty in condenser design [3]. Nevertheless, prior studies have shown these

⇑ Corresponding author at: Air Conditioning and Refrigeration Center (ACRC), Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, IL, USA. E-mail addresses: [email protected] (W.A. Davies III), [email protected] (P. Hrnjak). https://doi.org/10.1016/j.ijheatmasstransfer.2019.118666 0017-9310/Ó 2019 Elsevier Ltd. All rights reserved.

condensers to have a stratified flow regime [4], to have large variations in wall temperature and heat flux, and to have a HTC exceeding that of published correlations at low heat flux [5]. At higher heat flux, HTC has been shown [6] to follow the filmcondensation model of Dhir and Lienhard [7], which is based on the work of Nusselt [8]. 1.1. Correlations for heat transfer coefficient during condensation in stratified flow Condensation correlations are generally separated by applicability to annular flow or stratified flow [9]. As the flow in large flattened-tube condensers has been found to be predominantly stratified [10], only stratified-flow correlations will be discussed here. The majority of these correlations are based on the laminar film condensation analysis of Nusselt [8] for a vertical flat plate.

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Nomenclature A C cp D Dh dh f G g Ga

area (m2) empirical constant specific heat at constant pressure (J kg1 K1) diameter of tube (m) hydraulic diameter of tube (m) hydraulic depth; dh = Acs/Top Width (m) friction factor mass flux (kg m2 s1) gravity (m s2)
 galileo number; (Ga ¼ qf qf  qg gD3 =l2f )

H HTC i ifg J Ja k L LMTD _ m n Nu P P pwetted Pr Q q00 R Re Redh

height of tube (H = 0.216 m)(m) heat transfer coefficient (W m2 K1) specific enthalpy (J kg1) specific enthalpy of vaporization (J kg1) dimensionless superficial velocity Jakob number; Ja ¼ cp ðT s  T w Þ=ifg thermal conductivity (W m1 K1) length (m) log mean temperature difference (°C) mass flow rate (kg s1) number of water channels (n = 18) Nusselt number based on tube hydraulic diameter pressure (kPa) tube perimeter (m) tube perimeter covered by stratified condensate layer(m) Prandtl number heat transferred (W) heat flux (W m2) radius of curvature of semi-circular portion of tube (m) Reynolds number based on tube hydraulic diameter Reynolds number based on hydraulic depth of the stratified condensate layer temperature (°C) thickness (m) bulk water temperature (°C) condenser heat transfer coefficient (W m2 K1) uncertainty velocity (m s1) tube width (m) vapor quality Lockhart-Martinelli parameter axial position (m)

T t Tbulk U u v W x X Z

atm b bulk c c1

Subscripts A annular a air

For this case, the HTC is inversely proportional to the wall-steam temperature difference.

2




30:25

qf qf  qg g ifg L3 5 NuL ¼ 0:9434 k f lf ð T s  T w Þ

ð1Þ

To account for convection and subcooling in the film, Rohsenow [11] showed that a modified latent heat of condensation can be used, with increasing importance at high Jakob numbers.

  0 ifg ¼ ifg 1 þ 0:68Jaf

ð2Þ

Expanding on this analytical framework, Dhir and Lienhard [7] replaced the gravitational acceleration with an effective gravitational acceleration (gravitational acceleration in the direction of wall curvature) in order to apply the film condensation theory to

atmosphere bottom bulk condensate primary condenser (consisting of air and water-cooled test sections) c2 secondary condenser cs cross-section f fluid film condensing film region fo fluid only g gas go gas only i inlet int interfacial j channel number k index of position along water channel direction L length loss lost to the atmosphere lt Laminar-turbulent (relating to Lockhart Martinelli parameter) mean circumferentially-averaged o outlet r water-cooling loop rectangle rectangular portion of tube ri water-cooling-loop inlet ro water-cooling-loop outlet s steam sat saturation temperature semi-circle semi-circular portion of tube strat stratified liquid layer t top tt turbulent-turbulent (relating to the Lockhart-Martinelli parameter) w wall Greek symbols void fraction b proportion of tube perimeter in the condensing-film region D incremental step h angle around the tube circumference (radians) l viscosity (Pa s1) q density (kg m3) r surface tension (N m1) u inclination angle (°) U two-phase multiplier

a

arbitrary axisymmetric bodies. These analyses provide the basis for the majority of stratified-flow condensation correlations. Subsequent empirical correlations often add coefficients to Eq. (1) account for convection, tube geometry, and void fraction. Many also include an additional term accounting for single-phase convection in the stratified liquid layer. In an analytical correlation supported by experimental work, Chato [12] divided the stratified flow into two regions: the stratified liquid layer at the tube bottom, and the condensing-film region along the rest of the tube wall. He neglected the heat flux through the stratified liquid, and determined the average Nusselt number in a given tube cross-section to be:

2




30:25

qf qf  qg g i0fg D3 5 Nu ¼ 0:5554 kf lf ðT s  T w Þ

ð3Þ

3

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Chato also applied this correlation to inclined flow, which requires the solution of an open-channel-flow analysis for the depth of the condensate layer. He showed that this depth of condensate will decrease for low (u = 0–10°) downward tube inclinations, while the HTC through the condensing film will decrease by a factor (cosu)1/4. Rosson and Myers [13] measured local HTC during condensation of methanol and acetone in order to develop a correlation considering heat transferred through the stratified layer as well as through the film. They created a correlation in two parts: one for the condensing-film region and one for the stratified-liquid region. Their correlation in the condensing-film region uses an empirically-derived coefficient that accounts for the effect of vapor shear on the film:

2 Nufilm ¼

4 0:31Re0:12 g






30:25

qf qf  qg gi0fg D3 5 kf lf ðT s  T w Þ

ð4Þ

In the stratified liquid layer, they used an analytical correlation based on the von Karman analogy between heat and mass transfer:

pffiffiffiffiffiffiffiffiffiffi Uf ;lt 8Ref  Nustrat ¼  lnð1þ5Prf Þ 5 1þ Pr

ð5Þ

f

Uf ;lt ¼

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 12 1þ þ X lt X 2lt

ð6Þ

The circumferentially-averaged Nusselt number is the sum of these two components, weighted for the ratio of tube perimeter covered by each regime:

Nu ¼ bNufilm þ ð1  bÞNustrat

ð7Þ

b is an empirical correlation for the ratio of the tube circumference in the condensing film region to total tube circumference:

b ¼ Re0:1 for g b¼

0:5 Re0:6 g Ref

Ga

< 6:4
105

ð8Þ

0:5 Re0:6 1:74
105 Ga g Ref pffiffiffiffiffiffiffiffiffiffiffiffiffiffi for > 6:4
105 Ga Reg Ref

2




30:25

qf qf  qg gifg D3 4 5 Nu ¼ kf lf ðT s  T w Þ 1 þ 1:11X 0:58 tt

  þ 1  hf =p Nustrat ð10Þ

hf ¼ angle from top of tube to top of stratified liquid 0:4 Nustrat ¼ 0:0195Re0:8 f Pr f Uf ðX tt Þ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c Uf ðX tt Þ ¼ 1:376 þ c12 X tt

Nu ¼

Nufilm Dh þ ð2p  hÞDNustrat 2p D 2




ð13Þ 30:25

qf qf  qg gifg D3 5 Nufilm ¼ 0:7284 k f lf ð T s  T w Þ

ð14Þ

Pr 0:5 Nustrat ¼ :003Re0:74 f f f int

ð15Þ

f int

 11=4 0
1=2

qf  qg gt2c vg G @ A ¼1þ Gstrat vf r

ð16Þ

They assume constant depth of the condensate, tc, in the stratified liquid layer. Cavallini et al. [16] divided their correlation into DTwindependent and DTw-dependent regimes. The DTw-dependent regime encompasses all gravity-dependent flow regimes, which is broader than the stratified-only specification of the other correlations discussed here. Therefore, their correlation includes an additional annular-flow term, with weighting based on a dimensionless gas velocity, Jg:

h i  0:8  Nu ¼ NuA Jg;transition =J g  Nustrat J g =J g;transition þ Nustrat

ð11Þ ð12Þ

ð17Þ

Jg,transition is the dimensionless gas velocity at which the regime transitions from DTw-independence to DTw-dependence.  2
30:25 n o1 qf qf  qg gifg D3 0:3321 4 5 Nustrat ¼ 0:725 1 þ 0:741½ð1  xÞ=x kf lf ðT s  T w Þ   þ 1  x0:087 Nufo

ð9Þ

This combined analytical and empirical correlation fits well with their experimental data, although they admit significant experimental uncertainty in determining HTC at the bottom of the tube. Jaster and Kosky [14] used experiments with steam to make a correlation similar to that of Chato [12], neglecting the heat flux through the stratified layer. Dobson and Chato [9] considered heat flux through both the film and the stratified layer, and divided their correlation similarly to Rosson and Myers [13], although they did not measure local HTC. Their correlation is:

0:23Re0:12 go

The constants C depend on the liquid Froude number. Thome et al. [15] used a broad range of experimental data for refrigerants in round tubes in order to develop a flow regime-based correlation. The formulation of their stratified-flow correlation is similar to that of Dobson and Chato, although with an additional coefficient based on interfacial friction factor on the Nustrat term:

ð18Þ 0:4 Nufo ¼ 0:023Re0:8 fo Pr f

2 NuA ¼ Nufo 41 þ 1:128x0:8170

ð19Þ

qf qg

!0:3685

lf lg

!0:2363 1

lg lf

!2:144

3 5 Pr0:100 f ð20Þ

Lips and Meyer [17] developed a hydrodynamic model for HTC in inclined stratified condensation. Through modeling, they showed that the curvature of the surface of the stratified liquid has a significant effect on the mean HTC. Their model solves for the precise shape of the stratified liquid layer and assumes onedimensional conduction through this liquid layer. In the condensing-film region, they used Fieg and Roetzel’s [18] analytical solution for film thickness in an inclined tube. As with Chato’s [12] correlation, the thickness of the liquid film increases with cos (u)1/4. Shah [19] developed a correlation for inclined condensation from a large database of experimental results. His correlation encompasses all flow regimes, with transitions based on dimensionless vapor velocity. Inclination is not expressed explicitly in the correlation, but it affects the flow regime transition criteria. Nusselt number is considered to be insensitive to inclination from 90° upward to 30° downward tube inclinations. For stratified flow, the correlation is:

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2
Nu ¼



31=3

q qf  qg gD3 5 lf 2

f 4 1:32Re1=3 f

ð21Þ

This defers from the other stratified-flow correlations in having no dependence on DTw. Ewim and Meyer [20] modified the Cavallini [16] correlation based on their experimental results for low mass flux (50– 200 kg m2 s1), stratified-flow condensation of R134a in an 8.38 mm ID tube. They found that the accuracy of the previouslydeveloped correlations in predicting their data decreased as the mass flux decreased. Their correlation modifies the exponents in Eq. (18) for G  100 kg m2 s1: ! D0:755 n

Nustrat ¼ 0:725

0:265

kf



þ 1x

 0:25

1 þ 0:741½ð1  xÞ=x

0:3321

o1

2






30:245

qf qf  qg gifg 5 kf lf ðT s  T w Þ

4

developed or verified for large, flattened-tube air-cooled condensers (ACC). In addition, despite the many methods of determining HTC through the stratified condensate layer, there is very little data available for HTC through this liquid layer. The work of Rosson and Myers [13] is a notable exception. An additional concern is that all except the correlation of Kroger [22] have been developed for relatively constant heat flux in the condensing-film region. For the flattened-tube air-cooled condenser, heat flux can vary by two orders of magnitude in this region. To address these concerns, this paper presents experimental results for heat transfer coefficient in a flattened-tube steam condenser with non-uniform heat flux and wall temperature. HTC is presented in the stratified condensate layer, in the condensingfilm region, and as a circumferential average. The results are compared to the aforementioned correlations. From these results, a new correlation for stratified condensation HTC in a flattenedtube ACC is developed and presented.

Nufo ð22Þ

Ewim and Meyer [21] further showed that the HTC is significantly affected by tube inclination angle, both in magnitude and relationship to wall-steam temperature difference. However, they have not yet added this effect to their correlation. The only correlation developed specifically for flattened-tube air-cooled condensers is an analytical solution developed by Kroger [22]. His correlation takes into account the varying heat flux caused by the air-side temperature glide in each condenser cross-section:

" Nu ¼ 0:9245

#0:333 D3h q2f gcosðuÞifg dZ

    lf m_ a cpa ðT s  T ai Þ 1  exp UdZH= m_ a cpa ð23Þ

In Eq. (23), H is the tube height, dZ is the incremental tube length, and Kroger recommends approximating U by the air-side HTC. In all, there are several well-formulated correlations for stratified-flow condensation. However, none of them have been

2. Facility 2.1. Overview A novel heat transfer facility is used to determine local heat transfer coefficient for a flattened-tube steam condenser with non-uniform heat flux and wall temperature. The facility and experimental method are described in complete detail in [6]. A schematic of the experimental setup is presented in Fig. 1. The setup contains two consecutive test sections: an air-cooled condenser tube of 5.7 m length, and a water-cooled condenser tube of 0.12 m length. At the outlet of the air-cooled section, the collected steam and condensate flow directly into the water-cooled test section. These tubes are connected directly via flange in order to assure that no disruption is made to the flow regime. Flow regime and depth of the stratified condensate layer are observed in an adiabatic visualization section located directly downstream of the water-cooled condenser. An additional visualization section is located directly upstream of the air-cooled test section. Visualization along the full condenser length was performed in previous

Inlet Heater Gate Valve

Condensate Drain

Secondary Condenser

Ejector Pump

Water Tank

Ejector

Water Tank

Mass Flowmeter

Boiler 1 Q = 24 kW

Boiler 2 Q = 45 kW

Condensate Pump

Condensate Receiver

Condensate Trap

Fig. 1. Schematic diagram of the condenser test facility.

Condensate Receiver

Pump

W.A. Davies III, P. Hrnjak / International Journal of Heat and Mass Transfer 146 (2020) 118666

experimental work [5,10]. The condensers are mounted on a hinged truss, allowing for lifting to the entire range of downward tube inclinations. At the outlet of the downstream visualization section, the remaining steam is sent to a secondary condenser to be condensed. The flow rate of this remaining liquid is measured to verify the quality in the water-cooled test section. Noncondensables are removed by the ejector loop. In the air-cooled test section, local HTC is determined by a calibrated model. The method and results for local HTC in the aircooled test section are discussed in a previous paper [23]. In the water-cooled test section, local HTC is determined by a combination of experiment and computational fluid dynamics (CFD). This section is a cross-flow heat exchanger – the same as in the air cooled section – with cooling water pumped upwards in one pass, in direct contact with the outer wall of the condenser tube. The heat flux and temperature profile of the cooling water are intentionally non-uniform, to mimic the conditions in an operating air-cooled condenser.

5

Fig. 3. Diagram of water-cooling loop.

2.2. Test tube The condenser tube in this experiment is 216 mm in inner height and 16 mm in inner width, as seen in Fig. 2. The tube is steel with aluminum cladding on the outside. In the air-cooled section, the tube has wavy aluminum fins. The fins are 200 mm
19 mm, with a thickness of 0.25 mm. The finned tube is installed in an air duct, with crossflowing air pulled upwards through the fins. The water-cooled tube has the same geometry, but the fins are replaced by water flow that fully replicates the temperature conditions of the air-cooled section.

2.3. Air-cooled test section Air-side capacity is determined in the air-cooled test section, along with steam temperature and pressure. Air inlet and outlet temperatures are measured at 0.5 m intervals along the test tube with T-type thermocouples. Temperature of the stratified condensate layer is also measured at Z = 2 m and Z = 4 m along the tube. Air velocity is measured by a handheld hot-wire anemometer at 496 points along the tube. Gauge pressure is measured at con-

Fig. 2. Condenser tube cross-section views. (a) air-cooled tube; (b) water-cooled tube (no fins).

Fig. 4. Diagram of water-cooled test section, with water jacket covering tube (prior to adding insulation).

Fig. 5. Polycarbonate water jacket, (a) External view of jacket, matching view in Fig. 4; (b) End view showing water inlet and space for tube; notches on sides are for thermocouple wires. (c) View of inside one half of the jacket. Recesses for Tr thermocouples can be seen as holes in the middle three channels. Nine channels articulate flow of water that is in direct contact with the steel tube.

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denser inlet and outlet with differential pressure sensors. Atmospheric pressure is measured locally with a barometer. 2.4. Water-cooled test section The water-cooled test section is cooled via a polycarbonate water jacket that was manufactured using fused deposition modeling. A diagram of the water-cooling loop is presented in Fig. 3. The water-cooled tube is attached via flange to the air-cooled section in to ensure continuous development of flow regime. A schematic of the test section is presented in Fig. 4. The water jacket is presented in Fig. 5. The water jacket and tube form a single-pass counter-flow heat exchanger. The water jacket channels are designed to mimic the large temperature glide on the air side in an air-cooled condenser [5]. The steam-side flow is in the stratified regime, so the condensation HTC depends on wall-steam temperature difference. Therefore, we fully mimic the ACC conditions in order to develop a useful HTC correlation.

Channel geometry and instrumentation are shown in Fig. 6. Water temperature is measured at the inlet and outlet to the jacket, as well as at six locations in each of the three middle channels, on each side of the condenser tube. The thermocouples along each channel are positioned in recesses directly above the water channels, in order to measure the water temperature at the top of the channel. We used CFD simulation to determine the mixing-cup temperature of the water flow, and results are compared to the temperature measurements. Details of the CFD simulation are presented in [6]. Similarly to the water temperature measurements, wall temperature is measured in six locations at each of the three middle channels on both sides of the tube. The thermocouples are embedded in the 1.7 mm-thick steel wall of the tube. The thermocouple beads are covered with Omegabond 101 high thermal conductivity epoxy (published thermal conductivity of 1.0 W m1 K1) before insertion to ensure proper thermal contact with the wall. In addition, the channels for the wires are filled with this thermal epoxy after insertion in order to affix the thermocouples, to maintain a

Fig. 6. Diagram of water-cooled section (pictured in Figs. 4 and 5); (a) top and end views; (b) side view. Wall (Twjk) and water (Trjk) temperatures are measured in the three center channels along the channel length. Polycarbonate (shown as gray in Fig. 4 and beige in Fig. 5) depicted in green for visibility.

Fig. 7. Diagram of condensate subcooling measurements in the two test sections. Two subcooling measurements, Tc1 and Tc2, are made in the air-cooled test section, and one, Tc3, is made at the condenser outlet.

7

W.A. Davies III, P. Hrnjak / International Journal of Heat and Mass Transfer 146 (2020) 118666

smooth wall and to ensure proper heat conduction through the wall. All temperatures are measured with T-type thermocouples.

Table 1 Operating conditions for water-cooled experiments. Parameter

2.5. Measurement of condensate subcooling

Range 2

Steam mass flux [kg m

To measure the condensate temperature, T-type thermocouple probes are placed with their tips 1 mm above the bottom condenser wall, in order to measure condensate temperature as close to the condenser wall as possible. These thermocouples were positioned at 2 m and 4 m along the tube length in the air-cooled section, as well as at the condenser outlet. A diagram of the condensate thermocouples (Tc) is shown in Fig. 7.

s

1

]

1

Steam mass flow rate [g s ] Vapor superficial Reynolds number [–] Liquid Reynolds number (based on hydraulic depth of condensate) [–] Mean heat flux [kW m2] Steam quality [–] Condensation pressure [kPa] Inlet water-steam temp. difference [°C] Inclination angle [°]

0.75– 5.3 2.5–18 0–7000 10– 4100 2–67 0–0.74 52–108 32–77 0–38

2.6. Experimental procedure The system is run with steam pressure above atmospheric at startup to expel all non-condensables from the system. When the temperature in the secondary condenser indicates that all noncondensables have been removed, the release valve is closed. Boiler power is then reduced in order to create a vacuum in the system. Due to the large size of the system, ingress of non-condensables at vacuum conditions is inevitable. These collect in the secondary condenser and are removed via a water-powered ejector. Quality in the water-cooled section is controlled by adjusting fan power in the air-cooled condenser and by adjusting system mass flow rate. Mass flow rate is controlled via boiler power. The flow rate of cooling water is controlled to maintain a constant water temperature difference between the inlet and the outlet. This control allows imitation of the temperature glide across the fins in the air-cooled condenser. Flow regime and depth of the stratified condensate layer are observed and measured via an adiabatic visualization window at the outlet of the water-cooled test section. 2.7. Experimental procedure: Heat transfer coefficient in the stratified condensate layer

Table 2 Uncertainty of instruments. Variable

Instrument

Uncertainty

T P _ m va

T-type thermocouple Diaphragm differential pressure transducer Coriolis mass flow meter Hot-wire anemometer

±0.1 °C ±0.25 kPa ±0.1% ±3%

Steam mass flux is the total steam mass flow rate divided by the inner cross-sectional area of the condenser tube:

Gs ¼

_s m Acs

Steam quality at the inlet to the water-cooled test section is determined by two methods. For the first method, condensate generated by the water-cooled section is subtracted from mass flow rate of condensate leaving the primary condenser. The result is divided by the system mass flow rate.

xr;method1 ¼ To determine HTC through the condensate at various depths, the valve at the outlet to the condenser is closed, and the condenser tube is flooded to the desired depth of condensate. The tube is kept at 0.5° inclination, to ensure a near-constant depth of condensate through the test section. Depth is defined as the height from the bottom of the tube to the point where the condensate contacts the tube wall. Depth of the condensate is measured via the visualization window directly downstream of the test section. Once the depth reaches the desired level, the outlet valve is opened just enough to maintain the condensate depth constant. Once the depth and the flow rate of condensate are constant, data are recorded. This method makes it possible to measure a wide range of condensate depths and velocities. 2.8. Test conditions The test conditions in Table 1 describe the conditions at the inlet of the water-cooled test section. Table 2 lists the uncertainties of the instruments.

ð25Þ

_ c1  Q r =ifg m _s m

ð26Þ

Capacity of the water-cooled test section, Qr, is determined by Eq. (28):

  _ r T ro cp;ro  T ri cp;ri þ Q loss Qr ¼ m

ð27Þ

Q loss ¼ U loss As LMTDratm

ð28Þ

The value of Qloss – the heat lost from the cooling water to the atmosphere – was about 5% of Qr for all tests. Uloss was determined independently by isolating the water-cooled condenser. Once isolated, an electric heater was installed in the condenser, and the condenser interior was filled with water. A comparison was made between the electrical power of the heater and the capacity of the cooling water (Qr) in order to determine Uloss. The second method to determine quality is by using the capacity of the air-cooled test section, measured via heat transferred to the cooling air (Eq. (30)):

xr;method2 ¼

Q a =ifg _s m

ð29Þ

3. Method 3.1. Data reduction: Heat transfer coefficient Total steam mass flow rate is the sum of the condensate mass flow rates leaving each of the water-cooled test section and the secondary condenser.

_ c1 þ m _ c2 _s¼m m

ð24Þ

Air-side capacity, Qa, is determined via air velocity and air-side temperature difference. A complete description of this determination is available in Davies and Hrnjak [23]. Final steam quality is determined by an uncertainty-weighted average, as described by Park et al. [24]:




xr ¼

1 umethod1 2



xr;method1 þ 1 umethod1 2




þu

1 umethod2 2 1

method2

2

 xr;method2

ð30Þ

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W.A. Davies III, P. Hrnjak / International Journal of Heat and Mass Transfer 146 (2020) 118666

Eq. (32) shows that heat flux of the water-cooled section is determined by water-cooled capacity divided by the total steamside heat transfer area of the water-cooled section, which totals 0.0547 m2:

q00r ¼

Qr As

ð31Þ

Local capacity of the water-cooled section is determined from change in local water temperature in each channel, as in Eq. (33):

Q r;j;k ¼

 _r m T bulk;r;j;kþ1 cpr;j;kþ1  T bulk;r;j;k cpr;j;k þ Q loss;j;k n

Q loss;j;k ¼ U loss As;k LMTDratm;j;k    T bulk;j;kþ1  T atm  T bulk;j;k  T atm   ¼ T atm Þ ðT ln Tbulk;j;kþ1T ð bulk;j;k atm Þ

ð32Þ ð33Þ



LMTDratm;j;k

ð34Þ

The local water mass flow rate is determined by assuming equivalent mass flow rate in each of the 18 channels (n = 18). This was achieved through careful design – maintaining higher pressure drop through the channels than in the header – and verified by comparing temperature measurements in adjacent and opposing channels [6]. The value of Qloss – the heat lost from the cooling water to the atmosphere – varied from <1% of Qr,j,k at the bottom of the tube to about 10% at the top of the tube (where the water temperature is highest). Local heat flux is then determined by the local water-side heat transfer area, with the subscripts j and k in Eqs. (32) and (36) denoting the channel number (1–9) and axial location along the channel (1–6) respectively:

q00r;j;k

Q r;j;k ¼ As;k

ð35Þ

The local condensation heat transfer coefficient is then determined by local heat flux and temperature difference between the steam and the wall, and then accounting for the heat transfer resistance of the wall (Eqs. (37) and (38)):

HTC j;k ¼

q00 r;j;k T s  T w;j;k

HTC s;j;k

1 tw ¼  hj;k kw

ð36Þ

1 ð37Þ

The HTC through adjacent channels is then averaged to get the final local HTC in each of six locations along the tube wall, from tube bottom to top (Fig. 6):

P3 HTC s;k ¼

j¼1 hs;j;k

3

P6

k¼1 HTC s;k As;k P6 k¼1 As;k

HTC mean ¼

ð41Þ

3.2. CFD method for determination of local bulk water temperature To determine Tbulk,r in Eq. (32), a two-dimensional CFD simulation is performed for one of the water channels. Inlet water temperature, water flow rate, and condenser wall temperature are input from the experiment. The simulation is validated by comparison with experimental measurements of local and outlet water temperatures. Complete details of the CFD simulation are provided in [6]. 3.3. Data reduction: Condensate depth, condensate velocity, and void fraction The receding contact angle of water on polycarbonate is about 80° [25], which means the condensate in the visualization section has a nearly-flat surface (see Figs. 8 and 9). This makes determination of the cross-sectional area of the condensate in the visualization section simple: the condensate has a semi-circular shape at the tube bottom, and a rectangular shape above the bottom.

Ac ¼ Asemicircle þ Arectangle

Asemicircle

ð42Þ

8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < R2 cos1 Rtc   ðR  t Þ 2Rt  t 2 c c c R ¼ : 2 pR =2 

Arectangle ¼

if t c < R

0 ðt c  RÞ  W

if t c < R

if t c < R if t c > R

ð43Þ

ð44Þ

The radius of curvature of the tube bottom, R, is 8 mm, and the tube width, W, is 16 mm. The hydraulic depth of the condensate is determined by the condensate cross-sectional area divided by the top width of the condensate:

dh ¼

Ac Wt

Wt ¼

ð45Þ

( qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 R2  ð R  t c Þ 2 t c < R W

tc > R

ð46Þ

In the test section, however, the condensate contacts the (rusted) steel tube wall, which has a receding contact angle close to 0° [26]. In this case, the precise shape of the condensate surface can be determined via the procedure described in [5]. However, the

ð38Þ

HTC in the stratified condensate layer is simply the HTC in the lower-most measurement section:

HTC strat ¼ HTC s;1

ð39Þ

HTC in the condensing-film region is the average of the HTC in the remaining measurement sections. The HTC over the unmeasured area at the very top of the tube is assumed to remain constant:

P6 HTC film ¼

k¼2 HTC s;k As;k P6 k¼2 As;k

ð40Þ

Circumferentially-averaged HTC is the area-averaged value of the six local HTC measurements. The HTC over the unmeasured area at the very top of the tube is assumed to remain constant:

Fig. 8. Flow regime and condensate depth are observed and measured in the visualization window directly downstream of the water-cooled test section.

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9

Fig. 9. Schematic of water-cooled test section and visualization section, along with shape of condensate in each section, and description of measurement location in test section.

difference in hydraulic depth and cross-sectional area between the two methods is less than 3% for all cases, so this surface curvature could be neglected. The void fraction is calculated by area ratio:

9% when hydraulic depth is greater than 7 mm. The uncertainty increases when hydraulic depth is shallower than 7 mm.

4. Results and discussion

A  Ac a ¼ cs Acs

ð47Þ

3.4. Uncertainty Uncertainty in local HTC is calculated using the method of Taylor and Kuyatt [22] for Eq. (37). This method takes the square root of the sum of the squares of the deviation caused by the uncertainty in each of the component measurements. The main component of the uncertainty of this local HTC is the uncertainty of the local capacity, as determined by the CFD. A secondary cause of the uncertainty is the measurement uncertainty of the temperature difference between the wall and steam. The relative magnitude of these two contributions varies with test conditions. For example, the uncertainty of each thermocouple is 0.1 °C, and the temperature difference at the tube bottom varies from 36.7 °C to 68.0 °C. However, uncertainty of local HTC is near 12% for all tests. Uncertainty of mean HTC, determined as the average of the local HTCs, is 6%. The uncertainty of the mean HTC is much lower because the mean heat flux is measured experimentally. This eliminates the added uncertainty of the CFD simulation that is present in the determination of local HTC. Uncertainty of the hydraulic depth of the stratified condensate layer has three main contributors: uncertainty in measuring the depth of the condensate layer (tc), uncertainty in measuring the tube dimensions, and uncertainty in approximating the crosssectional area of the condensate. The uncertainty of the condensate depth measurement is ±0.3 mm. The uncertainty of measuring the tube width is significant because the tube width increases as operating pressure increases due to the non-circular shape of the tube. This uncertainty is ±1 mm. The uncertainty in approximating the condensate area is 7%, although this increases when the depth of the condensate layer is less than 10 mm. For example, when the condensate depth is 5 mm, the uncertainty of the area is 28%. The final uncertainty of the hydraulic depth (dh) of condensate is

4.1. Organization of results and proposal of new correlations Sections 4.2–4.4 discuss heat transfer through the stratified condensate layer at the tube bottom. Section 4.2 describes the subcooling of the condensate layer. Section 4.3 presents results for heat transfer coefficient in the stratified condensate layer. Section 4.4 presents a new correlation for HTC in the stratified condensate layer. Sections 4.5 and 4.6 describe HTC in the condensing-film region, with 4.5 presenting results and 4.6 discussing a new correlation for HTC. Finally, sections 4.7 and 4.8 combine these previous discussions to present the results and model for circumferentially-averaged HTC for a given condenser cross section.

4.2. Condensate subcooling along the condenser length Fig. 10 shows condensate temperature in comparison to vapor temperature along the condenser length for a tube inclined at 0.5°. Condensate depth at the tube outlet is varied artificially by varying the opening of a valve at the outlet of the condenser. This variation in condensate depth is created solely to show the effect of condensate depth on subcooling and HTC of the condensate. The varying outlet depth of the condensate is not a reflection of varying condenser operating conditions, such as inclination or air-side flow rate. The results show that condensate subcooling increases along the condenser length, and increases as condensate depth increases. Subcooling is negligible at Z = 2 m except for the condition with condensate depth equal to 112 mm at the condenser outlet (the maximum depth tested). In this deepest case, the condensate fills nearly half of the condenser tube at the outlet. For the other conditions, subcooling is less than 1.5 °C at Z = 2 m. There is significant subcooling at the condenser outlet for all conditions, with subcooling at the outlet increasing as condensate depth increases. Increased subcooling indicates a decrease in temperature gradient

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Fig. 10. Condensate temperature along the condenser at several different condensate depths. Condensate depth is measured at the condenser inlet and outlet, and is plotted on the right ordinate.

at the tube bottom, which should decrease the conductive heat transfer rate. These measurements of condensate subcooling are not used in the development of the correlation for HTC, nor are they intended to be used for precise modeling of condensate subcooling. They are provided to give additional insight into condenser operation and performance. 4.3. Results for heat transfer coefficient in the stratified condensate layer Fig. 11 shows the heat transfer coefficient at the bottom of the tube (over the area highlighted in red and labeled ‘HTCstrat’ in Fig. 9). The local bulk heat transfer coefficient at the tube bottom decreases as depth of the condensate increases for a constant mass flow rate. The primary cause for the decreasing HTC is a decrease in temperature gradient, which is inversely proportional to the condensate depth (Fig. 10). The HTC in this region also exceeds that predicted by the correlation of Dobson and Chato [9]. The experiment also exceeds the lower-bound assumption of onedimensional conduction through the liquid layer. This conduction model neglects any convective heat transfer. Fig. 12 shows that HTC through the stratified layer increases as a function of the ratio between Prandtl number of the saturated liquid, Prf,sat (top of the condensate river) and Prandtl number of the liquid evaluated at the wall temperature, Prf,w. The Prandtl number evaluated at the wall temperature was as much as three

Fig. 11. HTC through the stratified condensate layer decreases as hydraulic depth of the condensate increases. For depths <5 mm, HTC encompasses both the filmcondensing and stratified-condensate regions.

times greater than the Prandtl number evaluated at saturation temperature for the experimental points. This ratio is inversely related to the subcooling of the liquid. Miropolski [27] suggested this ratio as a factor effecting single-phase convective heat transfer coefficient, and results show that it is a strong predictor of HTC. Fig. 13 shows that HTCstrat is not strongly predicted by the Reynolds number based on the hydraulic depth of the condensate, Redh. Sano and Tamai [28] showed that the transition Reynolds number based on channel height for open channel flow is 1700, with fully turbulent flow found at Redh = 1900. However, their results are based on wider channels, and they artificially created turbulence and observed the transition to laminar flow. Both of these factors will produce a lower transition Reynolds number than is observed in our study. As an alternative criterion, Hanks and Ruo [29] found that for constrained pipe flow with high aspect ratio, the transition Redh can be as high as 2800. From these two criteria, we can conclude that the data at Redh < 1700 are definitively in the laminar regime. In this regime, HTCstrat is clearly independent of Reynolds number. The data recorded at higher Redh, with 1900 < Redh < 2700, could be considered turbulent based on the criterion of Sano and Tamai [28] or laminar based on the criterion of Hanks and Ruo [29]. In the visualization section at the tube outlet, laminar flow with a smooth liquid surface was observed, so we consider this flow to be laminar as well. For this low-Reynolds-number laminar flow, there is no dependence of HTC on velocity of the condensate. Fig. 14 shows that the measured velocities are below 0.04 m s1, and HTC does not increase as velocity increases. This result is expected for convective

Fig. 12. HTC through the stratified layer is shown to increase as the ratio of liquid Prandtl numbers at saturation temperature and wall temperature increases.

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recorded in order to verify the repeatability of the correlation. The correlation predicts 80% of these points to within 20% error, with a mean absolute percent error of 16%. 4.5. Results for heat transfer coefficient in the condensing film region

Fig. 13. Heat transfer coefficient through the stratified condensate layer is not strongly predicted by Reynolds number based on the hydraulic depth of the condensate (Redh).

laminar flow heat transfer, where Nusselt number is a function of channel geometry only [30]. 4.4. Correlation for heat transfer coefficient in the stratified condensate layer The correlation for HTC in the stratified layer is modeled after the correlations by Miropolski [27] and Dobson and Chato [9]. The proposed correlation for HTC through the stratified condensate layer is presented in Eq. (49).

HTC strat ¼ 200 Pr3:4 f ;sat



0:36 0:42

Prf ;sat kf ;sat dh Prf ;w W dh

ð48Þ

The first factor (Prf,sat) indicates that HTC increases as thermal conductivity increases, or as fluid viscosity decreases. The second factor (Prf,sat/Prf,w) is an indicator of the liquid subcooling, to which it is inversely proportional. The third factor (dh/W) is the aspect ratio of the stratified layer. For laminar convective heat transfer, Nusselt number increases as this aspect ratio increases from one to infinity [31]. The correlation was developed for the range of conditions listed in Table 3. Comparison with experimental results, as well as with previous correlations and a one-dimensional conduction model is presented in Fig. 15. The correlations of Dobson and Chato [9], Rosson and Meyers [13], and the conduction model show no relation to the experimental data over the range of experimental conditions. The correlations, in particular, are insensitive to the hydraulic depth of the condensate. The new correlation predicts 61% of the data to within 20% error, with a mean absolute percent error (MAPE) of 21%. Fig. 15 also shows additional experimental points that were

This section presents results for the average HTC in the condensing film region, above the stratified condensate layer. In this region, the film is thin (40 um), so condensation is the predominant mode of heat transfer. Fig. 16 shows that HTC in this region is not affected by quality. Decreasing quality has two main effects in this condenser: vapor velocity decreases, and depth of the stratified condensate layer increases. Decreasing vapor velocity decreases the shear force on the liquid film. Vapor shear can increase HTC by creating waves on the liquid film, but in this condenser, vapor shear is too low to have any effect. Increasing depth of the condensate has no effect on the film-condensation region. This agrees with the model of Dhir and Lienhard but disagrees with the film-condensation models of Rosson and Myers [13] and Dobson and Chato [9]. Fig. 17 shows that HTC decreases as the wall-steam temperature difference increases. The increasing temperature difference increases the thickness of the condensate film, which decreases the HTC. The correlations shown in the figure all predict a power-law relationship between Tw,film and film HTC, with HTCfilm  T0.25 w,film. The experimental results show a relationship of HTCfilm  T0.55 w,film. Fig. 18 shows that film HTC increases as temperature glide of the cooling water (DTr) increases. This trend can be explained in two manners. An increased DTr is caused by decreased water flow rate [6]. This decreased flow rate decreases the heat flux, which increases the HTC. Additionally, as DTr increases, the average wall-steam temperature difference (DTw) decreases. This also leads to an increase in HTC, as seen in Fig. 17. Fig. 19 shows that the mean HTC decreases as the inlet temperature difference (DTri) increases. Increasing inlet temperature difference leads to increased heat flux. As heat flux increases, the rate of condensation increases, which increases the film thickness. The thicker film increases the resistance to heat conduction and decreases the HTC. Fig. 20 shows that HTC in the condensing-film region decreases as inclination angle increases. This agrees with the prediction of Chato [12]. In the condensing-film region, increasing inclination angle increases the falling path of the condensing film. This increases the mean film thickness, which decreases the mean HTC. According to the theory of Nusselt, the mean HTC is proportional to Length1/4, so HTC is proportional to (cosu)1/4. Fig. 21 shows that film HTC increases as steam temperature increases. This is predominantly the result of two changes in properties. Thermal conductivity of the liquid film increases as temperature increases, which increases conduction through the condensing film. Viscosity of the liquid film also increases as temperature increases, which decreases the thickness of the condensing film. Both of these changes have a positive effect on film HTC. Both the direction and magnitude of this change in HTC agrees with the model of Dhir and Lienhard [7].

Table 3 Range of variables used to develop the correlation for HTC in the stratified condensate layer.

Fig. 14. Heat transfer coefficient through the stratified condensate shows no dependence on velocity of the condensate.

Variable

Range

Ts [°C] Tw [°C] dh [mm] Redh [–]

91.7–104.0 30.4–57.7 13–132 810–4100

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Fig. 18. Heat transfer coefficient in the condensing-film region increases as air-side temperature glide increases.

Fig. 15. A comparison of experimentally-determined HTC through the stratified condensate layer with prediction by one-dimensional conduction and three correlations, including a new correlation proposed in this paper.

Fig. 19. Heat transfer coefficient in the condensing-film region decreases as inlet air-steam temperature difference increases. Fig. 16. Heat transfer coefficient in the condensing-film region does not vary with quality.

Fig. 20. Heat transfer coefficient in the condensing-film region decreases as inclination angle increases. Fig. 17. HTC in the condensing film region has a power-law increase as wall-steam temperature difference decreases.

4.6. Correlation for heat transfer coefficient in the condensing film region Based on the experimental data for HTC in the condensing-film region in the water-cooled condenser and in the air-cooled condenser from Davies and Hrnjak [23], a new correlation is proposed. In this condensing-film region, Nusselt-type condensation is assumed, but an extra factor is introduced to account for the tube inclination angle. The correlation requires input of the average wall

temperature of the entire tube cross-section. The correlation is fit to two-thirds of the data points, and validated with the remaining one-third of data points, to avoid overfitting and provide validation of the model. The range of vapor superficial Reynolds numbers included in the model development is 0–7000. The proposed correlation for film HTC is given in Eq. (50).

HTC film 0


 2 30:25

g q q  q i0 p3 cosu f f g fg 2 kf 4 5 ¼ 0:75 p=2 lf ðT s  T w Þkf

ifg ¼ ifg þ 0:68cp;f ðT s  T w Þ

ð49Þ

ð50Þ

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HTC film ¼ 4:5
104


kf 0:25 ðcosuÞ H

"

H3 q2f gifg dZ

    lf m_ a cpa ðT s  T ai Þ 1  exp U
dZ
H= m_ a cpa

#0:62

ð51Þ

This correlation (version 2) more accurately predicts the experimental data when the wall temperature is unknown, as seen in Fig. 24. 4.7. Results for circumferentially-averaged heat transfer coefficient

The correlation is compared to experimental results and three existing models in Fig. 22. The new correlation more-accurately captures the trend of the experimental data than do the existing models. Despite its accuracy, this correlation requires an accurate model of the wall temperature in order to be applied. To determine this wall temperature accurately, the condenser cross-section must be discretized into small sections, due to the varying heat flux and HTCs on the steam and air sides [32]. This discretization requires knowledge of the local air-side HTC. The local air-side HTC can usually be found only by performing CFD for the air side. Unfortunately, less computationally-expensive models can be significantly less accurate. For example, the authors created a simple condenser model by assuming constant air-side and steam-side HTCs to determine an average wall temperature in each cross section. Using this average wall temperature, the accuracy of the new correlation is drastically reduced, as shown in Fig. 23. The added uncertainty in determining the wall temperature profile makes the uncertainty of this correlation unacceptable. Therefore, if the wall temperature profile cannot be accurately predicted, an alternative correlation – without wall temperatue – is required. Kroger [22] developed such a correlation analytically for an ACC by assuming a constant air-side HTC (Eq. (23)). By fitting his analytical correlation to the current experimental data, an alternative correlation is proposed that does not require estimation of the wall temperature:

Mean (circumferentially-averaged) HTC is dominated by filmcondensation effects, because the majority of the wall is in the condensing-film region. For nearly all conditions, the void fraction is greater than 0.9, so the stratified condensate layer accounts for less than 10% of the area-averaged HTC. Therefore, the circumferentially-averaged HTC closely resembles the film HTC presented in section 4.5. Fig. 25 shows that mean HTC is not significantly affected by quality. This closely matches the results for the condensing-film region in Fig. 16. This contrasts with the prediction of Shah [19], but agrees with the prediction of Chato [12]. Although decreasing quality has no effect in the condensing-film region, it increases the depth of the stratified condensate layer. Increasing depth of the condensate decreases HTC through the tube bottom, however, the overall effect of quality is minimal – less than the uncertainty of the HTC measurements. Figs. 26 and 27 show the effects of heat flux and wall-steam temperature difference on the circumferentially-averaged HTC. These charts also compare the HTC determined in the watercooled section with the HTC determined in the air-cooled section, as explained in prior papers [5,23]. The results show that circumferentially-averaged HTC is a function of heat flux and wall temperature, with the strongest effect in the lower range of both variables. The general trend agrees with the predictions of Dobson and Chato [9] and Chato [12]. However, the experimental HTC exceeds that of the correlations when heat flux is less than 17 kW m2 or DTw is less than 10 °C for all of the experimental data sets. One possible explanation is the presence of dropwise condensation, which was observed during diabatic visualization in previous experiments in an air-cooled tube [5]. Figs. 28 and 29 show that circumferentially-averaged HTC increases as temperature glide of the cooling water (DTr) increases. This closely follows the trend of the condensing-film HTC, seen in

Fig. 22. Comparison of experimental data with three existing correlations, as well as with a new proposed correlation for HTC in the condensing-film region.

Fig. 23. Accuracy of the proposed correlation decreases significantly if the wall temperature is not accurately predicted.

Fig. 21. Heat transfer coefficient in the condensing-film region increases as steam temperature increases.

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Fig. 27. Circumferentially-averaged HTC decreases as wall-steam temperature difference increases.

Fig. 24. Comparison of experimental data for HTC in the condensing film region with several existing correlations, as well as a new correlation that does not required prior knowledge of the condenser wall temperature (Eq. (51)).

region, and therefore circumferentially-averaged HTC also decreases. This agrees with the correlation of Kroger [22]. Fig. 31 shows the effect of inclination angle on mean HTC. The HTC decreases slightly as inclination angle increases. This agrees with the predictions of Chato [12] and Kroger [22]. As discussed above, in the condensing-film region, increasing inclination angle decreases the HTC. A counteracting effect is a decrease in thickness of the stratified condensate layer as tube inclination increases [4]. This causes the HTC in the stratified layer to increase [6]. However, this increase only affects a small proportion of the condenser area, so the effect on mean HTC is negligible. Fig. 32 shows that circumferentially-averaged HTC increases as condensation temperature increases. This is predominantly the result of two changes in properties. Thermal conductivity of the liquid increases as temperature increases, which increases conduction through the condensing film and through the stratified liquid layer. Viscosity of the liquid also increases as temperature increases, which decreases the thickness of the condensing film and increases mixing in the stratified condensate layer. All of these changes have a positive effect on circumferentially-averaged HTC. Both the direction and magnitude of this change in HTC agrees with the prediction of Chato [12].

Fig. 25. Effect of quality on circumferentially-averaged HTC is negligible.

4.8. Correlation for circumferentially-averaged heat transfer coefficient The correlation for circumferentially-averaged HTC combines the correlations for the stratified condensate layer and the condensing-film region, as with the correlations of Dobson and Chato [9] and Rosson and Myers [13]. In the film region, Eq. (49) can be used if the average wall temperature can be estimated accurately. If a local model for wall temperature is not available, then

Fig. 26. Circumferentially-averaged HTC decreases as heat flux increases for all experimental data.

Fig. 18. These charts also show that the circumferentially-averaged HTC decreases as inclination increases. Fig. 30 shows that the circumferentially-averaged HTC decreases as the inlet temperature difference (DTri) increases. Increasing inlet temperature difference leads to increased heat flux. As heat flux increases, HTC decreases in the condensing-film

Fig. 28. Circumferentially-averaged HTC increases as temperature glide on the cooling side increases (Ps = 106 kPa).

W.A. Davies III, P. Hrnjak / International Journal of Heat and Mass Transfer 146 (2020) 118666

Fig. 29. Circumferentially-averaged HTC increases as temperature glide on the cooling side increases (Ps = 80 kPa).

Fig. 30. Circumferentially-averaged HTC decreases as inlet temperature difference increases.

15

Fig. 32. Mean HTC increases as condensation temperature increases.

To apply this correlation, an estimation of the depth of the stratified condensate layer is needed. The most accurate method is to use an open-channel-flow model, as proposed by Chato [12] or Davies and Hrnjak [33]. However, previous results have shown that for most qualities (x > 0.1), conventional round-tube correlations can accurately predict the void fraction, with the correlation of Thom [34] fitting the experiments most accurately. Fig. 33 shows that either void fraction model of Thom [34] or Davies and Hrnjak [33] is suitable for use with the model for quality >0.05. Below this quality, the void fraction model of Thom predicts condenser flooding, which causes a drastic decrease in HTC that was not seen experimentally. In order to avoid this error, if using one of the conventional void-fraction models designed for round tubes, assumption of constant void fraction when x < 0.05 will provide suitable results. Once the void fraction and the condensate cross-sectional area (Ac) are calculated, the wetted perimeter of the condensate can be estimated by the following equations [35]:

pwetted

8qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 > < W 2t þ 16 t2c Ac 6 pW 3 8 ¼
 2 2 > : pW þ 2 Ac pW =8 Ac > p W W 2 8

8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < 2 W 2  W  t 2 A 6 pW 2 c c 2 2 8 Wt ¼ : p W2 W Ac > 8 ( tc 

W 0:34

A0:68 c   Ac 1 p þ W  2 W 8

0:8

2

Ac 6 pW 8

2

Ac > p W 8

2

ð53Þ

ð54Þ

ð55Þ

Fig. 31. Circumferentially-averaged HTC decreases slightly as inclination increases.

Eq. (51) should be used in the condensing-film region. In the stratified liquid layer, Eq. (48) is used. The correlation is fit to twothirds of the data points, and validated with the remaining onethird of data points, to avoid overfitting and provide validation of the model. The correlation (Eq. (53)) is valid for condensation in a stratified flow regime in inclined tubes under conditions of non-uniform heat flux and wall temperature. The range of vapor superficial Reynolds numbers included in the model development is 0–7000, and liquid Reynolds number (based on hydraulic depth of the liquid) ranges from 800 to 4100.

HTC mean ¼





pwetted p HTC strat þ 1  wetted HTC film p p

ð52Þ

Fig. 33. Mean HTC predicted by the correlation using two different void fraction models: those of Davies and Hrnjak [33] and of Thom [34]. Predictions are compared to experimental results.

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Fig. 34 shows a comparison of several correlations for HTC with the new correlation (using the average wall temperature from the experiments). Of the previous correlations, that of Chato [12] and Cavallini et al. [16] are able to predict the experimental data most accurately. The new correlation is able to predict 77% of the data within 20%. Table 4 shows that the mean absolute percent error of the new correlation is 14% if using the void fraction correlation of Davies and Hrnjak [33], and 15% if using the void fraction correlation of Thom [34]. Among the other correlations for HTC, that of Chato [12] is by far the most accurate in predicting the current data, with a MAPE of 27%. Fig. 35 shows that if the wall temperature profile is unknown, the alternative correlation (Eq. (51)) provides nearly-equivalent accuracy to the correlation requiring input of wall temperature.

Table 4 Mean absolute percent error (MAPE) of predictions by several HTC correlations in comparison with experimental data from this paper and from Davies and Hrnjak [33]. HTC correlation

Void fraction correlation

MAPE

New Correlation New Correlation Chato [12] Rosson & Myers [13] Dobson & Chato [9] Cavallini et al. [16] Kroger [22] Shah [19]

Davies & Hrnjak Thom [34] Davies & Hrnjak Davies & Hrnjak Davies & Hrnjak Davies & Hrnjak Davies & Hrnjak Davies & Hrnjak

14% 15% 27% 318% 52% 27% 71% 64%

[33] [33] [33] [33] [33] [33] [33]

5. Summary and recommendations Experimental results for condensate subcooling and heat transfer coefficient have been presented. Heat transfer coefficient results have been divided into two regions: in the stratified condensate layer at the tube bottom and in the condensing-film region above this layer. HTC in the stratified layer has been shown to vary with depth and subcooling of the condensate, but not with velocity. HTC in the condensing-film region has been shown to vary with heat flux, wall-steam temperature difference, cooling-side temperature glide, inlet temperature difference, condensation temperature and inclination, but remain constant with changes in quality. These results have followed the general trends proposed by the Nusselt model for film condensation. Circumferentiallyaveraged HTC has been shown to closely follow the film condensation results, due to this region occupying the majority of the condenser area. In comparing these results to existing correlations, HTC correlations designed for round tubes have been unable to accurately predict the experimental data. To improve the prediction, a new correlation for circumferentially-averaged HTC during stratifiedflow condensation in large-diameter inclined, flattened tubes has been proposed. The new correlation accounts for heat flux through the stratified condensate layer as well as in the condensing-film region. Variations for use when the wall temperature is known

Fig. 35. If the wall temperature profile is unknown, the alternative correlation provides higher accuracy than existing models.

(Eq. (49)) and unknown (Eq. (51)) are presented. In addition, a simple void fraction correlation (Thom [34]) is shown to provide sufficient accuracy for use with the correlation. The correlation accurately predicts the experimental data both from the current

Fig. 34. (a) Comparison of experimental data (air-cooled and water-cooled) to previous correlations; (b) Comparison of experimental data (air-cooled and water-cooled) to previous and new correlations.

W.A. Davies III, P. Hrnjak / International Journal of Heat and Mass Transfer 146 (2020) 118666

paper with a water-cooled condenser and from a previous experiment with an air-cooled condenser. There are several additional considerations when applying this correlation to designing an air-cooled condenser. The correlation is presented here for a circumferentially-averaged HTC, because this is the standard practice in the literature. However, assuming a constant steam-side HTC will result in an inaccurate capacity prediction when modeling an air-cooled condenser [33]. This is the result of large variations in air-side temperature and heat transfer coefficient in each condenser cross section [5]. A local model for heat transfer in each condenser cross section is recommended for accurate capacity prediction. The authors recommend using the correlation presented here for HTC in the stratified condensate layer, and then using the model of Dhir and Lienhard [7] to predict the local HTC in the condensing film region. This most closely follows the local HTC found experimentally [6]. In particular, the HTC in the stratified condensate layer is the most important for condenser design, even though it occupies a small area of the condenser. In this region, the steam-side HTC is the lowest, and therefore the heat flux is controlled by the steam-side HTC. Errors in estimating HTC in this region will significantly affect the accuracy of the condenser model, while errors in the condensing-film region will only have a minor effect. By dividing the proposed correlation into two regions, the authors are providing the basis to create the local condenser model with the stratified portion of the HTC correlation. Declaration of Competing Interest None. Acknowledgements The authors acknowledge the technical support provided by the Air Conditioning and Refrigeration Center (ACRC) at the University of Illinois at Urbana-Champaign, United States, and by Creative Thermal Solutions, Inc., United States that provided the experimental apparatus, laboratory space and support. References [1] EPRI, Power Plant Cooling System Overview: Guidance for Researchers and Technology Developers, Palo Alto, CA, 2015. [2] V. Dorjets, Many newer power plants have cooling systems that reuse water, in: Today in Energy, epub, Energy Information Agency, 2014. [3] A.J. Mahvi, A.S. Rattner, J. Lin, S. Garimella, Challenges in predicting steam-side pressure drop and heat transfer in air-cooled power plant condensers, Appl. Therm. Eng. 133 (2018) 396–406. [4] Y. Kang, W.A. Davies III, P. Hrnjak, A.M. Jacobi, Effect of inclination on pressure drop and flow regimes in large flattened-tube steam condensers, Appl. Therm. Eng. 123 (2017) 498–513. [5] W.A. Davies III, Y. Kang, P. Hrnjak, A.M. Jacobi, Heat transfer and flow regimes in large flattened-tube steam condensers, Appl. Therm. Eng. 148 (2019) 722– 733. [6] W.A. Davies III, P. Hrnjak, Local heat transfer coefficient during stratified flow in large, flattened-tube steam condensers with non-uniform heat flux and wall temperature, Int. J. Heat Mass Transfer 146 (2020). [7] V. Dhir, J. Lienhard, Laminar film condensation on plane and axisymmetric bodies in nonuniform gravity, J. Heat Transfer 93 (1971) 97–100.

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