Heat transfer correlation for saturated flow boiling of water - NSFC

Applied Thermal Engineering 76 (2015) 147e156

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Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng

Research paper

Heat transfer correlation for saturated flow boiling of water Xiande Fang*, Zhanru Zhou, Hao Wang Institute of Air Conditioning and Refrigeration, Nanjing University of Aeronautics and Astronautics, 29 Yudao St., Nanjing 210016, China

h i g h l i g h t s Compiles a database of 1055 data points of H2O flow boiling heat transfer. Evaluates 41 correlations of flow boiling heat transfer coefficient. Generalize approach for developing experiment-based correlation. Propose a correlation of H2O flow boiling heat transfer in small channels. The new correlation has a mean absolute deviation of 10.1% for the database.

a r t i c l e i n f o

a b s t r a c t

Article history: Received 8 July 2014 Accepted 10 November 2014 Available online 18 November 2014

The saturated flow boiling heat transfer of water (H2O, R718) is encountered in many applications such as compact heat exchangers and electronic cooling, for which an accurate correlation of evaporative heat transfer coefficients is necessary. A number of correlations for two-phase flow boiling heat transfer coefficients were proposed. However, their prediction accuracies for H2O are not satisfactory. This work compiles an H2O database of 1055 experimental data points from micro/mini-channels from nine independent studies, evaluates 41 existing correlations to provide a clue for developing a better correlation of saturated flow boiling heat transfer coefficients for H2O, and then proposes a new one. The new correlation incorporates a newly proposed dimensionless number and makes great progress in prediction accuracy. It has a mean absolute deviation of 10.1%, predicting 81.9% of the entire database within ±15% and 91.2% within ±20%, far better than the best existing one. Besides, it also works well for several other working fluids, such as R22, R134a, R410A and NH3 (ammonia, R717), being the best for R22, R410A and NH3 so far. © 2014 Elsevier Ltd. All rights reserved.

Keywords: Water H2O Flow boiling Heat transfer Coefficient Correlation

1. Introduction The saturated flow boiling heat transfer of water (H2O, R718) has many applications, such as compact heat exchangers and electronic cooling. The calculation of H2O flow boiling heat transfer coefficients is important for designing such facilities. A number of correlations for two-phase flow boiling heat transfer coefficient have been proposed, and their applicability to H2O is an interest issue. Many studies assessed the applicability to H2O of the correlations of two-phase flow boiling heat transfer coefficients. Sumith et al. [52] investigated experimentally the characteristics of H2O flow boiling heat transfer in a 1.45 mm inner diameter

* Corresponding author. Tel./fax: þ86 25 8489 6381. E-mail address: [email protected] (X. Fang). http://dx.doi.org/10.1016/j.applthermaleng.2014.11.024 1359-4311/© 2014 Elsevier Ltd. All rights reserved.

(ID) vertical tube at atmospheric pressure, with mass flux from 23.4 to 152.7 kg/m2s, heat flux from 36 to 391 kW/m2, and quality up to 0.6. They examined the effects of mass flux, heat flux and quality on the flow boiling heat transfer coefficient and compared the measurements with flow boiling heat transfer correlations of [4,30] and [38]. It was found that the liquid film evaporation was the predominant heat transfer mechanism, that slug-annular and annular flow patterns were dominated, and that the three correlations largely under-predicted the heat transfer coefficient, especially for a low heat flux condition. The underprediction gradually decreased with increasing heat flux. Steinke and Kandlikar [47] preformed an experimental investigation of H2O flow boiling heat transfer at the atmospheric pressure in six parallel horizontal copper micro-channels with a hydraulic diameter of 207 mm in the range of mass flux from 157 to 1782 kg/ m2s, heat flux from 55 to 898 kW/m2, and vapor quality up to 0.958. It was observed that the local flow boiling heat transfer coefficient

148

X. Fang et al. / Applied Thermal Engineering 76 (2015) 147e156

Nomenclature

x

Bd Bo Cov D Fa Fr G h hlg M Nu p Pr PR q Re T We X

Greek symbols l thermal conductivity (W/m K) m viscosity (kg/s m) r density (kg/m3) s surface tension (N/m)

bond number boiling number convection number diameter, hydraulic diameter (m) Fang number Froude number mass flux (kg/m2 s) heat transfer coefficient (W/m2 K) latent heat of vaporization (J/kg) molecular mass (kg/k mol) Nusselt number pressure (Pa) Prandtl number reduced pressure heat flux from tube wall to fluid (W/m2) Reynolds number temperature ( C) Weber number Martinelli parameter

exhibited a decreasing trend with increasing quality. The comparison of the measurements with the [22] correlation showed good agreement for x > 0.2. For x < 0.2, the [22] correlation showed large underprediction, and the underprediction increased with decreasing quality. Wen et al. [62] conducted an experimental study of H2O flow boiling heat transfer at the atmospheric pressure in a vertical rectangular tube of 2 mm by 1 mm with heat flux ranging from 27 to 160 kW/m2, mass flux from 134 to 211 kg/m2s, and quality up to 0.3. They compared the measurements with 11 flow boiling heat transfer correlations [4,8,22,25,26,32,33,38,56,60,64]. It was shown that the [38] correlation had the smallest mean absolute deviation (MAD) of 28%, followed by the [25] correlation of 45%, the [4] of 46%, and the [64] of 47%. They thought that the conventional methods were unreliable and that it was in need to develop better correlations. Qu and Mudawar [42,43] tested H2O flow boiling heat transfer in rectangular channel heat sink containing 21 parallel 231 712 mm channels with mass flux from 135 to 402 kg/m2s, heat flux from 53.6 to 519.2 kW/m2, quality up to 0.17, and outlet pressure of 1.17 bar. With the experimental data, they evaluated 11 flow boiling heat transfer correlations [4,17,22,32,33,38,45,48,56,60,64]. The results showed that the [64] correlation provided best predictions with an MAD of 19.3% but did not capture the correct trend of heat transfer coefficient with vapor quality, while the [60] correlation provided a closer prediction of the trend but had a greater MAD of 25.4%. They pointed a need for new predictive tools that could both capture the correct micro-channel heat transfer trends and yield more accurate predictions. Diaz and Schmidt [9] conducted an experimental investigation of H2O flow boiling heat transfer at the atmospheric pressure in a 0.3 12.7 mm rectangular channel with mass flux from 200 to 500 kg/m2s, heat flux from 90.4 to 359.8 kW/m2, and quality up to 0.32. They compared the experimental data with the predictions of six correlations [4,23,32,38,48,65]. The results showed that the measurements were largely overpredicted at higher quality and remarkably underpredicted when quality less than 0.05 by all the correlations, and that no correlation could capture the trend of heat transfer coefficient with vapor quality, indicating a strong need for new predictive methods.

vapor quality

Subscripts crit critical point exp experimental f fluid g saturated vapor l saturated liquid lo liquid only, assuming all fluid as liquid pred predicted sat saturated tp two-phase tt turbulent liquid/turbulent gas w channel inner wall surface

Kuznetsov and Shamirzaev [31] conducted the experiment of boiling heat transfer of H2O flow in rectangular parallel stainless steel microchannels with a size of 0.64 2.05 mm in crosssection and a typical wall roughness of 10e15 mm at atmospheric pressure. The local flow boiling heat transfer coefficients were measured at low mass flux of 17 and 51 kg/m2s, heat flux from 30 to 150 kW/m2, and vapor quality up to 0.8. They compared the measurements with the correlations of [22] and [65] and observed that the two correlations demonstrated incorrect trend of heat transfer coefficient with vapor quality at smaller mass flux. Kim and Mudawar [27,28] evaluated 13 previous correlations of saturated flow boiling heat transfer [1,3,8,11,17,32,37, 38,40,45,56,60,64] with a database for flow boiling in mini/microchannels, among which there are 485 data points from experiments with H2O. The [3] correlation performed best for the H2O data, with an MAD of 23.9%. They proposed a new generalized correlation for pre-dryout flow boiling in mini/micro-channels, which had an MAD of 21.2% for the H2O data. The accuracies of the [27] and [3] correlations were quite good, but the data points for H2O were limited. Therefore, their accuracies to H2O need to be confirmed using a large database. The above brief review clearly shows that the applicability of existing correlations of flow boiling heat transfer coefficients to H2O remains unclear. Results from different authors are not consistent. One reason for this is that only very limited data were used. Most authors conducted the evaluations only with their own measurements. Kim and Mudawar [27] used H2O data from multiple sources, but only 485 data points were compiled. Another reason for this is that the correlations evaluated were limited. The maximum number of the correlations involved was only 13, while the existing correlations of flow boiling heat transfer coefficients are more than 40. It is also clearly demonstrated that there is a need to develop a more accurate correlation for H2O flow boiling heat transfer. Flow boiling heat transfer depends on working fluids. People have been studying flow boiling heat transfer intensively for more than 50 years, and a correlation that works satisfactorily for a majority of working fluids has not been found yet. Therefore, it is necessary to develop a correlation specific for H2O.


This work aims at developing an accurate correlation of heat transfer coefficient of saturated flow boiling of H2O. A database containing 1055 data points of saturated flow boiling heat transfer of H2O from nine independent laboratories around the world is compiled, with which 41 correlations of saturated flow boiling heat transfer coefficient are evaluated and analyzed. The purpose of using H2O experimental database to evaluate correlations that were not developed for H2O is to find better existing correlations and to provide clue to develop new one for predicting H2O flow boiling heat transfer. Based on the evaluation results and the H2O database, a new correlation of saturated flow boiling heat transfer coefficient for H2O is proposed. The new correlation greatly boosts the level of the prediction accuracy for H2O and also works well for some other working fluids.

149

Table 2 Experimental uncertainty of data sources. Data source

T

p

[52] [47] [46] [62] [31] [16]

±0.1 C ±0.1 C ±0.2 C e ±0.2 C ±0.1e0.8%

e ±0.69 kPa e ±50 Pa 3.75 kPa ±0.2e13.6%

[42,43] [9] [2]

±0.3 C ±0.1 C ±0.5 C

±3.5% ±0.5 kPa ±1%

q (%)

G (%)

x (%)

h (%)

±1% e ±1.5 ±5 ±3 ±3.3b ±1.2c e e ±4.7

e e ±1.8 ±6 e ±6.2b ±2.8c ±4 ±0.1a ±2

e e e e e ±16.6b ±16.7c e e e

e ±8.6 e d

e ±13.8b ±13.5c e e ±14.8

e Not mentioned. a For full scale of the instrument. b For channel with 198 241 mm. c For channel with 378 471 mm. d From 20% at a superheat of 1.5 K to 8% at a superheat of 5 K.

2. Experimental data of H2O flow boiling heat transfer coefficients From nine independent sources around the world, 1055 data points of heat transfer coefficients of saturated flow boiling of H2O are obtained, as listed in Table 1. The experimental parameters cover the ranges of mass flux from 17 to 1782 kg/m2s, heat flux from 27.7 to 4788 kW/m2, vapor quality from 0.0001 to 0.958, hydraulic diameter from 0.207 to 1.73 mm, and saturation pressure from 1.01 to 16 bar. The experimental uncertainties of data sources are listed in Table 2. From the criterion stated in Ref. [23] and the common expression in the field, all the data are from micro/mini-channels. Among the database, 606 (57.4%) data points were from horizontal channels, and the rest were from vertical upward flow. Figs. 1e3 show the experimental data point distributions. Fig. 1 illustrates the distribution of the Reynolds number Re with the hydraulic diameter Dh. The liquid Reynolds number Rel varies from 4.9 to 2554.5, and the vapor Reynolds number Reg varies from 7.9 to 7447.5. There are 83.6% of the data having Rel within 100e1100, 8.3% within 4.9e100, and 8.1% greater than 1100. There are 83.1% of the data having Reg within 200e4000, 10.1% within 7.9e200, and 6.8% greater than 4000. Fig. 2 shows the liquid-vapor flow

distribution of the 1055 data points. Taking Re 1000 as laminar (viscous) flow and Re 2000 as turbulent flow, 22% of the data are in laminar liquid-turbulent vapor region, 44.5% in laminar liquidlaminar vapor region, 0.9% in turbulent liquid-laminar vapor region, and none in turbulent liquid-turbulent vapor region. Fig. 3 depicts the data distribution of vapor quality, from which it can be seen that most data have quality less than 0.2, accounting for 79.5%.

3. Applicability of existing correlations of saturated flow boiling heat transfer coefficient to H2O 3.1. Existing correlations of saturated flow boiling heat transfer coefficient Totally 41 existing correlations of saturated flow boiling heat transfer coefficient are evaluated based on the H2O database of 1055 data point, including nine CO2-specific ones [6,7,11,12,41,54,55,57,63] and 32 others [3,4,8,13,17e22,24e27,29,30,32,36e39,44,45,49e51,53,56,60,61,64,

Table 1 Experimental data sources of H2O flow boiling heat transfer. Data source

Parameter range: Tsat ( C)/psat (bar)/G (kg/m2s)/q (kW/m2)/x

Geometry range: Dh (mm)/La (mm)/εb (mm)/Wc (mm)/Hd (mm)/Flow direction/Channel type

[52]

100/1.01/23.4e152.7/36e391/0.003e0.604

[47]

100/1.01/157e1782/55e898/0.008e0.958

[46]

100/1.01/200e1500/35e196/0e0.228

[62]

100/1.01/134e211/27.7e122/0e0.22

[31]

102.8/1.12/17e51/35e150/0.01e0.75

[16]

100/1.01/340e1373/458.5e4788/0e0.47

[42]; [43]

104.1/1.17/135e402/53.6e519.2/0e0.17

[9]

99.6/1.01/200e500/90.4e359.8/0e0.327

[2]

120.21e201.31/2e16/100/50e160/0.023e0.593

1.45/100/e/e/e/Vertical upward/Single circular stainless steel tube 0.207/57.15/e/0.214/0.2/Horizontal/Slightly trapezoidal parallel copper channels 0.48/300/e/0.807/0.346/Vertical upward/Rectangular cooper tube 1.33/248/e/2/1/Vertical upward/Rectangular stainless steel tube 0.975/120/10e15/2.050/640/Vertical upward/Rectangular parallel stainless steel channels ①0.217/21.9/e/0.241/0.198; ②0.419/21.9/e/0.471/0.378/Horizontal/Single rectangular channel 0.349/44.8/1/0.713/0.231/Horizontal/Parallel rectangular copper channels 0.586/200/e/12.7/0.3/Vertical upward/Single rectangular nickel alloy inconel 600 tube 1.73/300/e/e/e/Horizontal/Single circular stainless steel tube

a b c d e

Channel length. Channel roughness. Channel width. Channel height. Not applicable

Number of data points 85 190 36 36 84 132

220 208 64

150


65]. Readers interested in knowing the forms of the correlations may want to refer [15] and other sources. The mean absolute deviation (MAD) is taken as the criterion to gauge prediction accuracy, and the mean relative deviation (MRD) is used to check whether a correlation has over-prediction or under-prediction.

N yðiÞ 1 X pred yðiÞexp MAD ¼ N i¼1 yðiÞexp MRD ¼

N yðiÞ 1 X pred yðiÞexp N i¼1 yðiÞexp

(1)

(2)

3.2. Overall assessment

Fig. 1. Distribution of channel size with Reynolds numbers.

Fig. 2. Laminar/turbulent flow distribution of the experimental data.

The overall assessment of the 41 existing correlations of saturated flow boiling heat transfer coefficient is listed in Tables 3e5. Table 3 shows that there are 10 correlations having an MAD < 40%. The [13] and [12] correlations perform best, with the MAD of 25.4% and 30.6%, respectively. The [13] correlation has the smallest MAD not only for the entire database, but also for the horizontal channels and vertical channels, respectively. The rest eight correlations having an MAD < 40% are [53,36,63,27,37,32,50,26], with the MAD of 32.5%, 34.9%, 35.1%, 35.3%, 37.1%, 39.1%, 39.7%, and 39.8%, respectively. All the top 10 correlations have similar MAD for both horizontal channels and vertical channels except those of [63] and [37]. The R134a database for developing the [13] correlation has reduced pressure PR (the ratio of the saturation pressure to the pressure at the critical point) from 0.075 to 0.32, the CO2 database for developing the [12] correlation has PR from 0.136 to 0.908, while 94% of the data in the present H2O database only have PR of 0.005, one order smaller than those of the R134a database and the CO2 database. However, the correlations of [13] and [12] can roughly predict this part of the H2O database. The mechanism of the effect of PR on flow boiling heat transfer coefficient remains unclear and needs to be investigated. Table 4 shows that there are five correlations having 40 MAD < 50%, which are [3,19,60,38,8], with the MAD of 40.2%, 40.2%, 45.5%, 47.2%, and 49.2%, respectively.

Fig. 3. Vapor quality distribution of the experimental data.


151

Table 3 Prediction deviation of correlations with MAD < 40% against the database (%). Flow direction

Deviation

[13]

[12]

[53]

[36]

[63]

[27]

[37]

[32]

[50]

[26]

Horizontal

MAD MRD MAD MRD MAD MRD

27.3 0.6 22.9 8.0 25.4 3.0

31.5 20.1 29.4 12.3 30.6 16.8

33.2 5.3 31.5 15.8 32.5 9.8

37.5 29.1 31.4 4.8 34.9 14.7

29.7 5.8 42.3 2.9 35.1 4.6

35.0 11.0 35.8 9.7 35.3 10.4

42.4 30.2 29.9 7.4 37.1 14.2

40.3 4.9 37.6 18.9 39.1 5.2

41.7 36.2 37.1 33.9 39.7 35.2

41.7 7.8 37.2 16.8 39.8 2.6

Vertical upward Total data

Table 5 shows the prediction deviation of the top four existing correlations against each data source. It can be seen that, the top three correlations have the largest MAD and large over-predictions for the [2] data, whose pressures are 2e16 bar, much larger than the rest data sources.

3.3. Variation of prediction of the top four correlations with quality The dependence of predictions of the top four correlations on quality is illustrated in Fig. 4. The experimental data used are from the [52] 1.45 mm ID tube in Fig. 4a, the [9] 0.3 12.7 mm (Dh ¼ 0.586 mm) rectangular channel in Fig. 4b, and the [31] 0.64 2.05 mm (Dh ¼ 0.975 mm) rectangular channels in Fig. 4c and d. Fig. 4 indicates the following: (1) The heat transfer coefficient of H2O two-phase flow boiling varies with vapor quality differently. Fig. 4a shows that the heat transfer coefficient increases moderately with increasing quality at lower quality, and decreases with increasing vapor quality at higher vapor quality. Fig. 4b shows that the heat transfer coefficient decreases dramatically with increasing vapor quality within quality less than 0.1. In Fig. 4c and d, the heat transfer coefficient decreases very slightly with increasing vapor quality, demonstrating a nucleate boiling dominant feature. The mechanism of effect of vapor quality on flow boiling heat transfer coefficient is not clear from the open literature. (2) The [13] and [12] correlations have the best ability to capture the trend of heat transfer coefficient with quality, but they may fail sometimes as shown in Fig. 4b. The [53] correlation is a nucleate type, whose prediction is independent of the vapor quality. The [36] correlation shows a slight decrease in heat transfer coefficient with increasing quality. Table 6 shows the deviation of the top four correlations for different vapor quality bands. In the database of 1055 experimental points, there are 307 (29.1%) data points in the range of 0 < x 0.05, 440 (41.7%) in the range of 0.05 < x 0.15, 177 (16.8%) in the range of 0.15 < x 0.3, and 123 (11.7%) in the range of 0.3 < x 0.7. Table 6 does not include the quality band of 0.7 < x 1 because only 8 data points are in this range. It can be seen that most (70.8%) of the data have x 0.15, suggesting that as a micro/min-channel heat sink, flow boiling of H2O may usually operates at very low vapor quality.

From Table 6 it can be seen that all the top four correlations underpredict the database by two digits from 17.9% to 34.1% in the range of 0 < x 0.05. This phenomenon was observed by several researchers [9,47]. The evaluation and analysis of the existing correlations above suggests a need for a new correlation that can predict more accurately the heat transfer coefficient for saturated flow boiling of H2O and capture the trend of heat transfer coefficient with quality. 4. Development of a new correlation 4.1. Approach for developing a new correlation Extensive computer tests have been conducted to develop a new correlation for coefficient of saturated flow boiling heat transfer of H2O based on the database of the 1055 data points. Through integrating and modifying the methods described in Refs. [5,12e14], the approach summarized below is used to develop the new correlation: (1) To collect dimensionless parameters from the correlations that have better performances. From the evaluation above, there are 10 correlations that have an MAD < 40% and five correlations that have 40% MAD < 50. The commonly used dimensionless numbers in the 15 correlations include the Nusselt number Nu, the liquid Reynolds number Rel, the liquid only Reynolds number Relo, the Boiling number Bo, the liquid Prandtl number Prl, the Fang number Fa, the liquid only Weber number Welo, the Bond number Bd, the liquid only Froude number Frlo, the convection number Cov, and the Martinelli parameter X. The vapor only Reynolds number, the confinement number (equivalent to Bd), and the vapor Reynolds number and the vapor Table 5 Prediction deviation of top four existing correlations against each data source (%). Data source

Deviation

[13]

[12]

[53]

[36]

[52]

MAD MRD MAD MRD MAD MRD MAD MRD MAD MRD MAD MRD MAD MRD MAD MRD MAD MRD MAD MRD

13.8 10.9 33.8 3.8 41.7 3.8 21.8 19.4 14.6 10.5 23.4 23.2 15.0 4.3 26.8 13.0 57.9 57.1 25.4 3.0

21.1 12.1 34.6 16.6 46.8 0.8 18.5 4.8 41.9 41.7 14.1 0.5 23.7 16.1 26.6 5.5 84.7 84.5 30.6 16.8

28.4 24.9 34.4 22.9 30.9 3.0 37.4 37.4 29.6 29.4 38.5 6.1 22.0 2.8 32.6 6.2 56.6 40.1 32.5 9.8

31.6 9.2 45.1 45.1 78.3 77.1 36.5 34.7 24.7 12.8 69.9 69.9 11.2 8.4 25.1 7.6 38.0 31.5 34.9 14.7

[47] [46] [62] [31]

Table 4 Prediction deviation of correlations with 40% MAD < 50% against the database (%). Flow direction

Deviation

[3]

[19]

[60]

[38]

[8]

Horizontal

MAD MRD MAD MRD MAD MRD

40.8 39.7 39.2 38.9 40.2 39.4

44.4 28.9 34.6 14.4 40.2 10.5

41.8 39.7 50.4 35.1 45.5 37.7

42.1 13.6 54.1 10.1 47.2 12.1

50.1 47.9 48.0 47.7 49.2 47.8

Vertical upward Total data

[16] [42,43] [9] [2] Total data

152


Fig. 4. Effect of vapor quality on flow boiling heat transfer coefficient: predictions vs. measurements.

Weber number also appear in the 15 correlations. The dimensionless parameters of the physical or operation parameter ratio in the 15 correlations include vapor quality x, reduced pressure PR, density ratio (rl/rg), and the viscosity ratio (ml,f/ml,w), where ml,f and ml,w are the liquid dynamic viscosity valued at the fluid temperature and the inner wall surface temperature, respectively.

Nu ¼ htp

D ll

ð1 xÞGD Rel ¼ ml Relo ¼

GDh ml

cp;l ml Prl ¼ ll q Bo ¼ Ghlg rl rg s Fa ¼ G2 D g rl rg D2 Bd ¼ s

Welo ¼

Frlo ¼

G2 D h rl s

(10)

G2 gDr2l

(11)

(3) PR ¼ (4)

p pcrit

There are different forms for the convection number Cov, differing in the values of the constant exponents. So are the Martinelli parameter X. The commonly seen forms are as the following:

(5)

Cov ¼

1x x

(6) (7)

(12)

Xtt ¼

1x x

0:9 0:5 rg rl

!0:1 0:9 0:5 rg ml rl mg

(13)

(14)

(8)

(9)

(2) To take the top 10 correlations listed in Table 3 as tentative model forms, based on which computer tests with the methods of least squares and error analysis are performed to determine baseline forms.


153

Table 6 Deviation of the top four correlations for different vapor quality bands (%). Data source

x

a

[52]

[47]

[46]

[62]

[31]

[16]

[42,43]

[9]

[2]

Total data

[13]

b

( 0,0.05] (0.05,0.15] (0.15,0.3] (0.3,0.7] (0,0.05] (0.05,0.15] (0.15,0.3] (0.3,0.7] (0,0.05] (0.05,0.15] (0.15,0.3] (0.3,0.7] (0,0.05] (0.05,0.15] (0.15,0.3] (0.3,0.7] (0,0.05] (0.05,0.15] (0.15,0.3] (0.3,0.7] (0,0.05] (0.05,0.15] (0.15,0.3] (0.3,0.7] (0,0.05] (0.05,0.15] (0.15,0.3] (0.3,0.7] (0,0.05] (0.05,0.15] (0.15,0.3] (0.3,0.7] (0,0.05] (0.05,0.15] (0.15,0.3] (0.3,0.7] (0,0.05] (0.05,0.15] (0.15,0.3] (0.3,0.7]

[12]

[53]

[36]

MAD

MRD

MAD

MRD

MAD

MRD

MAD

MRD

22.5 12.9 5.2 3.2 43.2 16.2 29.1 60.7 39.2 35.2 84.6 e 33.3 8.6 10.6 e 20.5 12.3 13.9 14.5 39.0 22.1 9.1 17.2 28.3 8.7 17.7 e 38.9 11.6 34.1 63.8 51.7 33.8 57.9 68.5 35.5 14.0 26.0 37.9

20.9 10.6 1.5 0.8 43.2 9.0 26.4 60.7 38.1 34.2 84.6 e 33.3 7.1 10.6 e 20.5 10.9 11.1 8.6 39.0 22.1 9.1 13.4 28.3 3.5 17.7 e 38.9 2.7 34.1 63.8 35.1 33.8 57.9 68.5 34.1 2.6 21.7 33.1

19.9 18.6 27.6 21.7 32.1 15.6 44.7 70.4 40.8 43.6 100.7 e 22.3 11.1 29.3 e 73.7 53.6 57.1 33.2 20.1 7.7 16.9 21.4 14.1 25.7 41.1 e 24.0 19.2 55.6 81.8 74.4 69.5 88.8 88.7 24.2 22.1 47.2 53.6

0.3 12.9 27.2 21.7 32.1 8.5 44.7 70.4 38.0 43.6 100.7 e 21.9 11.1 29.3 e 73.7 53.6 57.1 32.9 20.1 1.7 16.9 21.4 12.7 25.7 41.1 e 23.9 18.3 55.6 81.8 69.8 69.5 88.8 88.7 19.3 19.6 47.1 53.5

27.4 35.8 22.5 17.0 49.0 30.4 32.8 27.3 39.6 13.4 40.2 e 35.8 39.4 37.5 e 44.5 33.9 31.7 28.3 43.9 32.1 36.0 66.4 28.8 17.6 33.3 e 36.3 22.1 50.4 96.9 81.9 65.3 52.4 53.8 37.3 26.0 37.8 36.8

21.5 32.2 21.9 16.3 49.0 20.2 10.3 10.6 2.4 3.3 40.2 e 35.8 39.4 37.5 e 44.5 33.9 31.7 28.3 43.6 7.2 21.2 60.2 28.8 4.3 31.2 e 36.1 6.7 43.4 96.9 65.0 58.2 38.9 30.9 32.7 4.6 11.2 1.7

49.6 18.8 20.5 35.8 59.3 44.0 37.9 37.4 62.1 97.2 116.2 e 54.5 17.7 9.7 e 17.8 42.1 17.3 22.6 77.1 70.3 65.4 53.7 17.6 9.1 6.0 e 32.4 16.6 26.7 53.3 61.8 70.0 40.5 20.1 43.8 30.2 35.3 29.3

45.8 0.2 17.8 35.8 59.3 44.0 37.9 37.4 60.1 97.2 116.2 e 54.5 17.2 9.7 e 17.8 19.5 17.3 22.6 77.1 70.3 65.4 53.7 17.6 5.3 0.9 e 31.6 4.8 26.7 53.3 61.8 70.0 40.2 5.1 17.9 11.3 11.9 22.0

-Not applicable. a The number in the bracket is not covered. b The number is covered.

(3) The forms that show remarkable merits in the above process are chosen as the baseline forms. At this stage, the correlations of [13] and [12] demonstrate remarkable advantages over the others among the top 10. The [13] and [12] correlations have similar form, which can be generalized as the following:

htp ¼ Nu

ll D

(15)

. Nu ¼ 0:00061ðS þ F ÞRel Fa0:11 Prl0:4 ln U

(16)

S ¼ f ðBoÞ

(17)

F ¼ f ðCov; Fa; Rel Þ ! ml;f U¼f ml;w

(18) (19)

For example, for the [13] correlation based on the R134a database of 2286 experimental data points, it follows

S¼

30000Bo1:13 36

F¼

x 0:95 r l 1x rg

U¼

1:023ml;f ml;w

Bo < 0:0026 Bo 0:0026

(20)

!0:4 (21)

(22)

(4) Through error analysis, the baseline model form is modified by adding and/or removing some dimensionless numbers collected in step (1), and then computer tests with the methods of least squares and error analysis are repeated to improve the baseline form. The dimensionless parameters given by Eqs. (4)e(14), vapor quality x, density ratio rl/rg, and viscosity ratio ml,f/ml,w have all been added to the S, F and U functions defined by Eqs. (17)e(19), being bested one by one. The constants in the convection number Cov have been adjusted to get better results.

154


(5) Step (4) is repeated until the MAD cannot be reduced at the MRD around zero. 4.2. New correlation The new correlation is of the form.

htp ¼ Nu

ll D

(23)

. Nu ¼ 0:00061ðS þ F ÞRel Fa0:11 Prl0:4 ln U

(24)

where

S ¼ 142:5Bo

0:9

M

0:55

x 0:9 r l F¼x 1x rg

U ¼ 1:02

ml;f ml;w

rl rg

!0:33 (25)

!0:35 (26) Fig. 5. New correlation: predictions vs. experimental data.

! (27)

where M is the molecular mass in kg/kmol. All physical properties of H2O are obtained using the REFPROP of NIST by Ref. [34]. For the database of the 1055 data points, the new correlation has an MAD of 10.1% and an MRD of 0.05%, predicting 81.9% of the entire database within ±15% and 91.2% within ±20%, while the best existing one [13] has an MAD of 25.4%, predicting 37.6% within ±15% and 49.3% within ±20%. Fig. 5 shows the comparison between the predictions of the new correlation and the experimental database, from which it can be seen that the predictions with the new correlation agree with the measured data very well. Also, the new correlation captures the trend of heat transfer coefficient with vapor quality well (Fig. 6). Hence, the new correlation is much better than the best existing one.

0 to 1, saturation pressure from 270 to 2560 kPa, and channel diameter from 0.5 to 8 mm [35], the new correlation has an MAD of 24.7% and an MRD of 19.2%, being about the same as the best existing one which also has an MAD of 24.7%. For the NH3 database containing 1157 experimental data points of flow boiling heat transfer with mass flux from 10 to 600 kg/m2s, heat flux from 2.0 to 240 kW/m2, vapor quality from 0.002 to 0.997, saturation pressure from 0.19 to 1.6 MPa, and tube ID from 1.224 to 32 mm [58], the new correlation has an MAD of 32.2% and an MRD of 12.7%, much better than the best existing one which has an MAD of 40.9%. The new correlation does not predict well for the CO2 database containing 2956 experimental data points [15] and the N2 database containing 1043 experimental data points [59].

4.3. The applicability of the new correlation to some other working fluids The new correlation also works well for R22, R134a, R410A and NH3, and its applicability to other working fluids needs to be checked. For the R22 database containing 1669 experimental data points of flow boiling heat transfer with mass flux from 49.6 to 742 kg/ m2s, heat flux from 1.9 to 57.5 kW/m2, vapor quality from 0.006 to 0.982, saturation temperature from 15.65 to 35 C, saturation pressure from 101 to 1355 kPa, and tube ID from 1.5 to 13.84 mm [10], the new correlation has an MAD of 18.4% and an MRD of 4.5%, far better than the best existing one which has an MAD of 32.7%. For the R134a database containing 2286 experimental data points of flow boiling heat transfer with mass flux from 42 to 1500 kg/m2s, heat flux from 1 to 165 kW/m2, vapor quality from 0 to 1, saturation pressure from 300 to 1300 kPa, and channel diameter (hydraulic diameter for rectangular channels) from 0.19 to 8 mm [13], the new correlation has an MAD of 18.8% and an MRD of 9.3%, placing the second after the [13] correlation which has an MAD of 14.2%. For the R410A database containing 1268 experimental data points of flow boiling heat transfer with mass flux from 100 to 1079 kg/m2s, heat flux from 5 to 38.5 kW/m2, vapor quality from

Fig. 6. Trend of heat transfer coefficient with quality: new model predictions vs. experimental data.


5. Conclusions (1) The database consisting of 1055 data points of saturated flow boiling heat transfer of H2O in micro/mini-channels are compiled from nine independent sources, which covers the parameter range of mass flux from 17 to 1782 kg/m2s, heat flux from 27.7 to 4788 kW/m2, vapor quality from 0.0001 to 0.958, saturation pressure from 1.01 to 16 bar, and hydraulic diameter from 0.207 to 1.73 mm. (2) An approach for developing experiment-based correlation is generalized based on the author's previous studies. Using the approach and the H2O database, a new correlation for saturated flow boiling heat transfer of H2O is proposed. It has an MAD of 10.1% and predicts 81.9% of the entire database within ±15% and 91.2% within ±20%, while the best existing one has an MAD of 25.4%, predicting 37.6% within ±15% and 49.3% within ±20%. (3) The new correlation also works well for R22, R134a, R410A and NH3, with an MAD of 18.4%, 18.8%, 24.7% and 32.2%, respectively, being the best one for R22, R410A and NH3 so far, while its applicability to other working fluids needs to be checked. (4) Based on the H2O database, 41 existing correlations of flow boiling heat transfer coefficient are evaluated, including nine CO2-specific ones and 32 others. There are 10 correlations having the MAD < 40%. The top two are [13] and [12], which have the smallest MAD of 25.4% and 30.6%, respectively, and also have the best ability to capture the trend of heat transfer coefficient with quality. The rest eight are [53,36,63,27,37,32,50,26], with the MAD of 32.5%, 34.9%, 35.1%, 35.3%, 37.1%, 39.1%, 39.7%, and 39.8%, respectively. (5) The purpose of using the H2O database to evaluate the existing correlations is to find the better correlations and to provide a clue to develop new one for predicting H2O flow boiling heat transfer. The evaluation results cannot be construed as ranking the correlations. A number of the evaluated correlations do not work for H2O, which may be due to they were not developed for H2O. On the other hand, a correlation that is better than another one for the H2O database may not be as good as the same counterpart for some other databases. (6) The heat transfer coefficient of H2O flow boiling varies with quality in many ways. The mechanism behind this is not clear from the open literature. More efforts should be made to better understand the mechanism of H2O flow boiling heat transfer for developing more reliable prediction methods. (7) The reduced pressure of 94% data points in the H2O database is one order smaller than those of the R134a database for developing the [13] correlation and the CO2 database for developing the [12] correlation. However, the correlations of [13] and [12] can roughly predict this part of the H2O database, indicating that the mechanism of the effect of PR on flow boiling heat transfer coefficient remains unclear and needs to be investigated. Acknowledgments This study is supported by National Natural Science Foundation of China (51176074). References [1] B. Agostini, A. Bontemps, Vertical flow boiling of refrigerant R134a in small channels, Int. J. Heat. Fluid Flow. 26 (2005) 296e306.

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