Correlations for calculating heat transfer of hydrogen pool boiling

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Correlations for calculating heat transfer of hydrogen pool boiling Lei Wang a, Yanzhong Li a,b,*, Feini Zhang c, Fushou Xie a, Yuan Ma a a

Institute of Refrigerating and Cryogenic Engineering, Xi'an Jiaotong University, Xi'an 710049, China State Key Laboratory of Technologies in Space Cryogenic Propellants, Beijing 100028, China c Mechanical Science and Engineering, University of Ilinois, Urbana, IL 61801, USA b

article info

abstract

Article history:

Understanding the heat transfer characteristics of hydrogen pool boiling as well as con-

Received 11 July 2015

structing reliable correlations to guide the boiling heat transfer analysis is of significance to

Received in revised form

the applications of liquid hydrogen (LH2). In the present paper, the available hydrogen

17 June 2016

experimental data in literature are summarized and analyzed. Based on these data, several

Accepted 29 June 2016

existing correlations aiming at different boiling regimes are evaluated or modified in order

Available online 13 August 2016

to realize the quantitative analysis of hydrogen boiling heat transfer. After sufficient comparison studies, several improved correlations for nucleate boiling, critical heat flux

Keywords:

(CHF) and minimum heat flux (MHF) are proposed. Moreover, a complete set of correlations

Liquid hydrogen

for hydrogen boiling heat transfer are summarized and subsequently a predicted hydrogen

Heat transfer

boiling curve is constructed. The results show that heat flux in the nucleate boiling regime

Critical heat flux

is approximately a function of DT2.5. For the calculation of hydrogen film boiling, Breen &

Pool boiling

Westwater correlation seems to be appropriate for revealing the heat transfer character-

Empirical correlation

istics no matter what heater geometry is used. For the hydrogen boiling under 0.1 MPa condition, heat fluxes at onset of nucleate boiling (ONB), CHF and MHF are in the order of 10 W/m2, 105 W/m2 and 103 W/m2, respectively, and the corresponding wall superheats are approximately 0.1 K, 3 K and 4 K. In addition, pressure effect on hydrogen boiling heat transfer is investigated, and it shows pressure has a significant influence on MHF while a little effect on the heat fluxes in liquid natural convection and film boiling regimes. CHF of hydrogen increases with the increase of system pressure firstly and then decreased with further pressure increase. The maximum of CHF locates at about P/Pc ¼ 0.35. Generally, the present study is beneficial to understand the boiling heat transfer characteristics of hydrogen, and the summarized correlations could provide researchers with reliable mathematical tools to conduct the boiling heat transfer analysis of hydrogen systems. © 2016 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved.

* Corresponding author. Institute of Refrigerating and Cryogenic Engineering, Xi'an Jiaotong University, Xi'an 710049, China. Fax: þ86 29 82668789. E-mail address: [email protected] (Y. Li). http://dx.doi.org/10.1016/j.ijhydene.2016.06.254 0360-3199/© 2016 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved.


Introduction As an excellent propellant, LH2 has been widely used in aerospace field ever since the beginning of human space age, and will still play an important role in the future space activities. Moreover, hydrogen energy will occupy a prominent position in the future energy consumption structure. It was reported that the proportion of hydrogen to the total energy consumption in Japan will reach 20% by 2035 and then 40% by 2050 [1,2]. Besides the effect as a fuel, LH2 can also be used as a coolant in various applications such as large-scale superconducting magnet and cold neutron moderator material for a spallation neutron source [3]. Throughout the whole hydrogen chain including its production, transportation and utilization, numerous pool boiling heat transfer phenomena could be encountered. Understanding the heat transfer characteristics of hydrogen pool boiling and constructing predictive correlations to conduct quantitative analysis are of significance to the reliable design and successful operation of LH2 system. In the flourishing period of aerospace activities in 1950s and 60s, a great deal of effort had been devoted to the associated fundamental and technological researches, and LH2 boiling heat transfer was one of them. Some researchers conducted experimental investigation to obtain the boiling heat transfer data under different conditions, and also a series of correlations aiming at different boiling regimes were proposed to perform preliminary design of hydrogen heat transfer system. Class et al. [4] presented the experimental data obtained for the heat transfer of boiling hydrogen from a relatively large flat surface under a variety of conditions, and the hydrogen experimental data in previous literature were also analyzed. The results showed that these heat transfer data were not in close agreement, which might indicate that obtaining boiling curve of hydrogen faced great challenge due to lack of reliable experimental data. Graham et al. [5] investigated the influence of gravity on pool boiling heat transfer of hydrogen under subcritical and supercritical pressures. The results showed that acceleration had a certain influence on the incipience of nucleate boiling but did not remarkably affect the established nucleate boiling. Seader et al. [6] conducted an extensive survey on heat transfer to boiling cryogenic fluids including hydrogen, and the reviews of available theoretical approaches to predict boiling phenomena as well as experimental data were summarized. Brentari et al. [7] conducted a systematic study on the boiling heat transfer for four cryogenic fluids, including nitrogen, oxygen, hydrogen, and helium. For the hydrogen pool boiling, experimental data for eight sets of nucleate boiling and three sets of film boiling were used to assess the accuracy of several famous correlations. The results suggested that in the nucleate boiling regime, the nucleate heat transfer correlation and the maximum heat flux correlation proposed by Kutateladze [8] could be applied. In the film boiling regime, Breen and Westwater correlation [9] was recommended to calculate the stable film boiling heat transfer, and Zuber correlation [10] was suggested to predict MHF. Bewilogua et al. [11] investigated the dependence of cryogenic boiling heat transfer on pressure, where experiments on a horizontal plane surface immersed in liquid helium, LH2, and liquid nitrogen (LN2) in

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the pressure range of 0.03 < P/Pc < 0.9 were considered. The results showed that the maximum of CHF was found at about 0.35 Pc, and Kutateladze correlation could be used to predict CHF of hydrogen. Louie and Steward [12] conducted an experimental investigation to reveal the transient heat transfer characteristic of LH2 pool boiling. The results illustrated that CHF of hydrogen was approximately 80 kW/m2. Merte [13] presented the experimental results of incipient and steady boiling of cryogenic liquids, both LN2 and LH2, under reduced gravity conditions. The influence of several parameters, including fluid used, heater surface temperature, heater geometry, heater orientation, and gravity condition, on boiling heat transfer were analyzed. In recent years, motivated by the needs of hightemperature superconductivity and aerospace explorations, the problems associated with heat transfer of hydrogen boiling had been paid attention once again, and several sets of experimental investigations had been conducted to reveal the heat transfer characteristics of hydrogen under various conditions. Shirai et al. [14] applied an experimental approach to investigate the heat transfer from a flat plate facing upward immersed in LH2 pool. The flat plate heater was 10 mm in width, 100 mm in length and 0.1 mm in thickness, and the heat transfer was obtained for the pressures from atmospheric pressure to 1.1 MPa. The experimental data showed that the maximum CHF of hydrogen locates at about 0.3 MPa. Subsequently, Shiotsu et al. [15] studied the transient heat transfer characteristics of LH2 pool boiling under saturated and subcooled conditions, and a flat plate heater with 5 mm in width, 60 mm in length and 0.5 mm in thickness was adopted. The results indicated that transient CHF was higher for higher subcooling conditions. Tatsumoto et al. [16] presented the experimental studies on heat transfer from a horizontal wire immersed in both liquid and supercritical hydrogen. The wire test heater was made of PtCo with a length of 101.8 mm and a diameter of 1.2 mm. The results showed that heat transfer coefficient in nucleate boiling regime was higher for higher pressure condition, and CHF was highest in the vicinity of 0.4 MPa which could be expressed by Kutateladze correlation. To obtain the character of hydrogen boiling heat transfer in low-gravity condition, Garcia [17] placed a horizontal disk heater with a diameter of 11.28 mm into a huge magnetic field, and different gravities were achieved by adjusting the magnetic field intensity. The experimental results showed that under low gravity conditions, both of nucleate boiling and film boiling heat transfer were lower than that in normal gravity. Wang et al. [18] proposed a scaling analysis to link the LH2 boiling heat transfers under different gravities. Through this analysis, the heat flux at any gravity level could be obtained if the data in the similar condition were available at reference gravity. A sufficient precooling operation is usually necessary for the application of cryogenic liquids including LH2, and different heat transfer mechanisms, including film boiling, transition boiling, and nucleate boiling, dominate the temperature decrease process successively. Existing researches showed that film boiling plays a dominant role in the temperature decrease process. The ratios of temperature decrease occupied by film boiling to the total temperature decrease were approximately 95.5% and 83.3% for LH2 and LN2

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precooling events, respectively [13]. Therefore, a great deal of effort had been devoted to the problems associated with cryogenic film boiling, and a series of well-designed experimental investigations had been performed to investigate the heat transfer characteristics of cryogenic liquids on different geometry heaters. Class et al. [4] briefly discussed LH2 boiling heat transfer data from a large flat surface including film boiling data, and the influences of heater orientation, heater surface condition, and system pressure were considered. The cryogenic film boiling from other heater geometries had also been experimentally investigated. Pomerantz [19] used a horizontal tube with 4.78 mm outer diameter as the heater to investigate the pool film boiling heat transfer under increased gravity conditions. Sakurai et al. [20] presented the experimental data of pool boiling heat transfer from horizontal tube heaters, and these data covered the situations of various liquids for wide ranges of pressure, liquid subcooling, surface superheat and tube diameter. Horie et al. [21] conducted an experimental study to investigate the properties of LH2 film boiling heat transfer, and a PtCo wire with a diameter of 1.2 mm was used as the electric heater. Merte and Clark [22] conducted a boiling experiment to reveal the gravity influence on boiling heat transfer properties, and a 25.4 mm diameter sphere heater was used to evaporate saturated LN2. Hsu and Westwater [23] measured the film boiling heat transfer properties at vertical tubes, and stainless steel tubes with outer diameters varied from 9.5 to 19.0 mm and lengths from 66.0 to 165.1 mm were used. Besides the experimental investigations, a series of predictive correlations had also been developed. Bromley [24] firstly proposed a theory to calculate the heat transfer coefficient in stable film boiling from a horizontal tube, and the associated equations were derived from a few simple premises and well verified by extensive experimental data but no LH2 data. For the prediction of cryogenic film boiling at a wire heater, the correlation proposed by Breen & Westwater had good predictive accuracy, which had been verified by Kida et al. [25]. Besides, Baumeister and Hamill [26] also derived a predictive expression on heat transfer coefficient of pool film boiling from horizontal wires, and comparison studies showed this expression agreed well with all available data for wire diameters varied from 0.0053 mm to 46.0 mm. Simoneau and Baumeister [27] indicated that Bromley correlation could provide good prediction for the film boiling heat transfer from cylinder heaters of 6.4e19.1 mm diameter. For the film boiling at horizontal wire heaters, it was found the heat transfer coefficient of boiling nitrogen increased as wire diameters decreased. For the film boiling from sphere heaters, Frederking and Clark [28], Tou and Tso [29], and Farahat and Nasr [30] had made their contributions, and different predictive correlations aiming at different boiling conditions had been proposed. Considering the similarity of film boiling and natural convection at vertical surface heaters, an analogy analysis was applied by Chu [31] to propose a heat transfer correlation for saturated film boiling. Comparison study showed this correlation agreed well with test data for water film boiling at vertical surfaces. It was also indicated that this correlation could be used for cryogenic fluids predictions. Berenson [32] discussed Taylor-Helmholtz hydrodynamic instability and its significance with regard to film boiling heat transfer from a horizontal surface, and an

analytical expression for predicting heat transfer coefficient of film boiling from horizontal surface was derived. As has been introduced above, LH2 has a wide prospect of application in future energy and aerospace fields, and researchers have been concerning about the heat transfer characteristics of hydrogen pool boiling for over fifty years. Though heat transfer correlations for different pool boiling regimes have been obtained by U.S. National Bureau of Standards in 1965 [7], which could be used to guide the design of hydrogen systems, it is still necessary to evaluate these correlations with more experimental data to strengthen the confidence in utilizing them or modifying them based on more sufficient comparisons. In the present paper, the pool boiling heat transfer of para-hydrogen is investigated, and the associated thermodynamic physics properties are derived from NIST [33]. The heat transfer correlations for different regimes are evaluated with the experimental data from literature, and a series of improved heat transfer correlations, which could be used to construct the boiling curve of hydrogen, are proposed. It is believed that the improved correlations can well predict the heat transfer characteristics of hydrogen pool boiling, and can supply researchers with reliable mathematical tools in designing the heat transfer system of hydrogen.

The boiling curve Fig. 1 displays a typical boiling curve for an independently controlled surface temperature, which is helpful to understand the heat transfer regimes of pool boiling. With the increase of DT, different heat transfer mechanisms occur successively at the heater surface. At very low wall superheat, no vapor bubble can be produced at the heater surface and heat is transferred from the wall to the surrounding liquid by natural convection. Relatively, the heat transfer coefficient associated with liquid natural convection is low. When DT reaches a certain value at which some nucleation sites on the surface are initiated, ONB condition occurs. After the wall superheat of DTONB, the portion of the boiling curve enters into the nucleate boiling regime, and any further increase in DT will cause more and more bubbles produced. Correspondingly, heat flux increases significantly with the increase of DT until the peak value point of CHF is reached. In this peak condition, portions of the surface are dry and covered with a vapor film. If the wall temperature increases beyond CHF, another regime which is usually called the transition boiling regime occurs. Within this regime, the heat transfer at the surface is unstable. The dry portions covered with vapor film and the rewet portions contacting with liquid may occur alternatively. Generally, the probability of dry surface portions increases with the rise of DT, and a decreasing mean heat flux is obtained. With the further increase of DT, another critical point, which usually referred to as Leidenfrost or MHF point, is reached. After that a stable vapor film could be sustained at the entire surface and the film boiling regime occurs. If the surface temperature is decreased gradually from a high value to the liquid temperature range, the system will progress through each of the regimes described above in reverse order. It has been stated that constructing boiling curve of hydrogen faces larger challenge than other liquids since the


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q Liquid natural convection

Nucleate boiling

Transition boiling

qCHF

Film boiling

qL

qONB

ΔT ONB

ΔT L

ΔT CHF

ΔT

Fig. 1 e Boiling curve regimes for an independently controlled surface temperature.

variation ranges of DT in the nucleate boiling and transition boiling regimes are much smaller than that of the other fluids [18]. For example, a typical nitrogen pool boiling at P ¼ 0.1 MPa experiences a 10 K nucleate boiling regime and a 25 K transition regime, while DT ranges of hydrogen at the nucleate boiling and transition boiling regimes are approximately 3 K and 1 K, respectively [13]. Such small ranges of DT require strict accuracy in measuring the temperature variation of boiling hydrogen process. This is also the reason why many experimental data on heat transfer of boiling hydrogen are not in close agreement [4]. Therefore, a larger number of reliable data on hydrogen boiling should be accumulated and more sufficient comparison analysis is suggested to reveal the heat transfer characteristics of hydrogen boiling.

Heat transfer correlations Liquid natural convection It is reported that the pool boiling behaviors of cryogenic fluids is generally consistent with that of non-cryogenic fluids [34]. Therefore, the proposed heat transfer correlations based on the experimental data of non-cryogenic fluids may be applied to analyze the heat transfer characteristics of cryogenic fluids. For the liquid natural convection heat transfer, the famous equation proposed by McAdams has been widely applied, as written below [35]. Nu ¼ C$ðGrPrÞn

(1)

For the natural convection heat transfer of LH2 and LN2, Coeling et al. [35] pointed out that both of turbulent or laminar heat transfers could occur at a flat plate surface, so that Eq. (1) with different values of C and n should be adopted under different conditions. For turbulent heat transfer, Coeling et al.

[35] recommended C ¼ 0.14 and n ¼ 1/3, and for laminar heat transfer, C ¼ 0.79 and n ¼ 1/4 were suggested. Shirai et al. [14] also studied the LH2 heat transfer from a horizontal plate heater, and the results of Eq. (1) with C ¼ 0.16 and n ¼ 1/3 were found to agree well with the test data. The natural convection heat transfer of LH2 from wire heater was investigated experimentally as well, and Eq. (1) with C ¼ 0.53 and n ¼ 1/4 could be used to reveal the heat transfer characteristics [16]. Fig. 2(a) displays the relation of Nu and GrPr for natural convection heat transfer of LH2, and the test data under different conditions are utilized to conduct the comparison analysis. It shows an observable difference in curve slopes represented by the test data could be observed, which may indicate different flow patterns dominate the heat transfer process under different ranges of (GrPr), and thus different values of C and n in Eq. (1) should be utilized. There is a slight difference between the results of Eq. (1) with C ¼ 0.16 and the results with C ¼ 0.14 for accounting for the turbulent heat transfer. Relatively, Eq. (1) with C ¼ 0.16 obtains a better agreement with the boiling data, and thus it is recommended in this paper to calculate the natural convection heat transfer of hydrogen from a flat plate heater. For the laminar heat transfer at a wire heater, Eq. (1) with C ¼ 0.53 and n ¼ 1/4 reaches a better agreement so that it is suggested in the calculation of LH2 heat transfer from wire heaters. Fig. 2(b) displays the comparison of calculated and experimental natural convection heat transfer of LH2 from a flat plate surface under P/Pc ¼ 0.56 condition. Although Eq. (1) with C ¼ 0.16 and n ¼ 1/3 somewhat underestimate the heat transfer intensity of LH2 natural convection, the overall variation trend of heat flux under the relatively high pressure condition could be given by this correlation. To maintain the consistency, Eq. (1) with C ¼ 0.16 and n ¼ 1/3 is insisted for the high pressure events. Fig. 2(c) displays the comparison results for heat transfer from a wire heater. It can be seen that Eq. (1) with C ¼ 0.53 and n ¼ 1/4 attains an excellent agreement.

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It can be seen that the heat transfer coefficient is independent of characteristic length, which is convenient for the heat transfer calculation. Based on the above analysis, the following equations could be suggested to conduct the heat transfer calculation of LH2 at different heater geometries. Laminar:

Nu ¼ 0:79$ðGrPrÞ1=4 Nu ¼ 0:53$ðGrPrÞ1=4

for for

flat plate heater wire heater

(3)

Turbulent: Nu ¼ 0:16$ðGrPrÞ1=3

(4)

Nucleate boiling In the nucleate boiling regime, Brentari et al. [7] pointed out that the nucleate heat transfer correlation proposed by Kutateladze could be applied, as shown below " 1=2 #3=5 1=2 h s s 4 qcpl rl ¼ 3:25 10 $ ll grl hfg rv ll grl #7=10 " #1=8 " 2 3=2 rl s P g grl ml ðsgrl Þ1=2

(5)

or 7=10 3=5 hlc r qcpl lc P ¼ 3:25 104 $ l $ $Ga1=8 $ pffiffiffiffiffiffiffiffiffi ll rv hfg ll sgrl

(6)

It shows that Kutateladze correlation is in terms of several dimensionless groups. Imitating the expression form in Ref. [17], another conventional expression of q as a function of DT could be derived by rearranging the parameters in Eq. (5), as shown below. q ¼ 5:66 1010 $

1:28 1:75 ll c1:5 P pl rl 1:5 m0:625 s0:9 h1:5 l fg rv

$DT2:5

(7)

Clarke [36] also proposed a correlation for the nucleate boiling heat transfer of cryogenic liquids. " 1=2 #2:89 cpl ðTw Tsat Þ T 1:8 q s ¼ 3:25 105 $ ml hfg gðrl rv Þ Tc hfg Pr1:8 l

Fig. 2 e Comparison of calculated and experimental natural convection heat transfer for LH2: (a) Nu vs. (GrPr); (b) q vs. DTl for flat surface heater; (c) q vs. DTl for wire heater.

Here, T/Tc term incorporates an additional pressure effect. T is the average value of the wall temperature and the liquid saturation temperature. Some scholars pointed out that it was difficult to obtain any general theoretical method of calculating heat transfer coefficient in nucleate boiling since the difficulty in determining nucleate sites. To predict the heat transfer coefficient relatively accurately, a simple type of equation was usually suggested as below [37] q ¼ a$DTb

Substituting n ¼ 1/3 and expressions of Nu and Gr into Eq. (1), the following correlation can be given, as shown below. 1=3 1=3 h$Lx gbl DTl L3x gbl DTl ¼C Prl 0h ¼ C$ll $ Prl 2 2 ll nl nl

(2)

(8)

(9)

where b ¼ 3 to 3.33. For different fluids, there are different values of the parameters a and b. For the heat transfer between LH2 and stainless steel wall, a ¼ 3285 and b ¼ 2.66 was suggested [38]. In the present study, Eq. (9) is also considered and the


experimental data in Ref. [14] are used to resolve these parameters. A careful comparison study suggests that the correlation with a ¼ 6309 and b ¼ 2.52 seems to have a good accuracy for hydrogen calculation. q ¼ 6309$DT2:52

(10)

It could be observed that the exponent value of DT in Eq. (10) is similar to that of Kutateladze correlation by Eq. (7). To validate the predictive abilities of the above correlations, a comparison study is conducted and the comparison results under different pressures are exhibited in Fig. 3. It shows that within the considered pressure range Eq. (10) has the optimal predictive accuracy, and all of its predicted results are in general agreement with the experimental data. In atmospheric pressure condition, Kutateladze correlation and Eq. (10) produce the similar predictions, both of which are generally agreement with the experimental data, while Clarke correlation underestimates the heat flux apparently. Moreover, it should be noted that both of Kutateladze correlation and Clarke correlation can account for the pressure effect on nucleate heat flux. Therefore, the variation of pressure exerts a certain influence on their predictions. By comparing the

17123

predicted results with experimental data, it is easy to observe the pressure effect on the predicted heat fluxes. In atmospheric pressure case, as shown in Fig. 3(a), Clarke correlation obviously underestimates the boiling heat transfer intensity, and in the maximum pressure case an apparent predicted deviation is obtained by Kutateladze correlation, as shown in Fig. 3(d). Relatively, the predicted results of Eq. (10) are generally agreement with the experimental data for all of the four cases though it does not involve the pressure effect. Therefore, it seems to conclude that within the considered pressure range, the system pressure has a little influence on the nucleate boiling heat flux of hydrogen, and Eq. (10) is suggested in the heat transfer analysis of hydrogen pool boiling due to its simple form as well as the better predictive accuracy.

Onset of nucleate boiling For the prediction of ONB, the following correlation has gained widespread acceptance [34]. qONB ¼

ll hfg rv 2 DT Pr2 8sTsat ONB l

(11)

Fig. 3 e Comparison of calculated and experimental heat flux in nucleate boiling regime under different pressure conditions.

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This correlation was found to agree well with experimental data for a wide variety of fluids, including cryogenic fluids. In the present paper, its applicability in LH2 is assessed as well. As has been indicated in Section (The boiling curve), ONB signifies the critical point of liquid natural convection and nucleate boiling regimes. Therefore, the heat flux at ONB should simultaneously satisfy the natural convection heat transfer correlation, that the heat transfer coefficient is calculated by Eq. (1), and the nucleate boiling heat transfer correlation, Eq. (10), at DTONB, and thus the following equation could be given. 2:52 qONB ¼ h$DTl ¼ 6309$DTONB

(12)

Then DTONB for hydrogen can be obtained as below. DTONB ¼

1 h$DTl 2:52 6309

(13)

For the heat transfer analysis of saturation LH2, DTONB ¼ DTl, and the wall superheat, DTONB, and the heat transfer coefficient, h, satisfy both of Eqs. (1) and (12). Hence, an iterative computation can be utilized to solve DTONB and h, and then the heat flux at ONB point can be obtained. For the heat transfer calculation in subcooling LH2, DTONB þ DTsub ¼ DTl is met, and together with Eqs. (1) and (12) the critical values at ONB can also be reached. Table 1 lists the comparison of experimental and calculated DTONB by Eqs. (11) and (13), respectively. It can be clearly demonstrated that both of Eqs. (11) and (13) produce significant predicted deviations, and Eq. (11) significantly underestimates DTONB for all of the considered cases and Eq. (13) generally overestimates DTONB. To develop a more accurate predictive correlation on ONB of hydrogen, a larger number of reliable hydrogen test data should be obtained in advance but not included in the present paper. Nevertheless, Eq. (13) is insisted in this paper to calculate ONB of hydrogen since Eqs. (1) and (10) are applied to calculate the heat transfers at natural convection and nucleate boiling regions, respectively, and Eq. (13) is derived on the basis of Eqs. (1) and (10).

Critical heat flux The following relation could be applied to resolve CHF [34]. qCHF;sat ¼ CK hfg rv

1=4 gsðrl rv Þ r2v

(14)

Based on the comparison study with CHF data of hydrogen pool boiling, Brentari et al. [7] and Bewilogua et al. [11] concluded that the correlation with CK ¼ 0.16 could give sufficiently accurate results for hydrogen.

Fig. 4 e Effect of pressure on CHF of hydrogen. In this paper, the experimental data in Refs. [11,14,16] are used to further evaluate the predictive ability of Eq. (14). Fig. 4(a) displays the relationship between pressure and the coefficient, CK, in Eq. (14). It can be seen that pressure has an apparent influence on CK, especially in high-pressure conditions in which a constant value of CK ¼ 0.16 may overestimate CHF. This prediction deviation had also been pointed out by Shirai et al. [14]. Therefore, an improved correlation for CK, which is a function of P/Pc, is proposed based on these experimental data. 5:68 P CK ¼ 0:18 0:14 Pc

(15)

Table 1 e Comparison of experimental and calculated DTONB for hydrogen. Heater geometries Wire [16] Flat surface [13] Flat surface [14]

Conditions P/Pc ¼ P/Pc ¼ P/Pc ¼ P/Pc ¼

0.54, DTsub ¼ 5.0 K 0.54, DTsub ¼ 8.3 K 0.078, DTsub ¼ 0 K 0.56, DTsub ¼ 8.56 K

Experimental DTONB

DTONB by Eq. (8)

DTONB by Eq. (10)

0.30 K 0.29 K 0.13 K 0.16 K

0.049 K 0.065 K 0.00012 K 0.056 K

0.91 K 1.07 K 0.076 K 0.98 K


Fig. 4(b) illustrates the comparison of experimental and calculated CHFs with the increase of pressure. It shows that the improved correlation could better reveal the variation of CHFs with pressure compared to the original one. CHF in subcooling LH2 had also been investigated by Shirai et al. [14] and Tatsumoto et al. [16], and the following equation was given. qCHF;sub qCHF;sat

0:8 cpl DTsub r ¼ 1 þ 0:065 v rl hfg

(16)

Compared to CHF in saturation liquid, subcooling liquid usually has a higher CHF. The ratio of CHF in subcooling liquid to that in saturation liquid is directly related to the liquid subcooling degree. Imitating the correlation form of Eq. (16), the subcooling effect on CHF of hydrogen is reconsidered in the present paper, and the relationship between CHF ratios and liquid subcooling terms is displayed in Fig. 5(a). It can be seen that with the increase of subcooling a larger ratio is observed, and the following correlation could be suggested. 0:8 cpl DTsub qCHF;sub r ¼ 1 þ 0:23 v qCHF;sat rl hfg

(17)

Fig. 5(b) shows the CHFs at pressures around 0.4 MPa, 0.7 MPa and 1.0 MPa versus subcooling of LH2. Compared to

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the original correlation the proposed correlation in Eq. (17) shows better accuracy.

Film boiling As has been introduced in Section (Introduction), film boiling heat transfer plays a dominant role in the cryogenic precooling operation, and a great deal of effort has been devoted to the problems associated with cryogenic film boiling heat transfer. To realize the quantitative analysis of film boiling heat transfer, a series of correlations have been proposed, and heater geometry effect is usually involved in constructing these correlations. Table 2 lists the representative correlations aiming at different heater geometries. Klimenko [40] proposed an approach to predict the pool film boiling heat transfer coefficient on a horizontal surface on the basis of Reynolds analogy since there was some similarity between film boiling and natural convection. Therefore, the following relation could be supposed to reveal the heat transfer characteristics of film boiling. NuLx ¼

" !#m h0fg hLx ¼ C1 RaLx $ lv cpv DT

(18)

After a careful comparison analysis it is found that all of the listed correlations in Table 2 could be rearranged as the correlation form of Eq. (18), and the differences of these correlations could be summarized as the different values of C1, Lx, h0fg , and m. By comparing the original formations and the unified formation, the parameters C1, Lx, h0fg , and m are also

Fig. 5 e Effect of liquid subcooling degree on CHF of hydrogen.

obtained and presented in Table 2. For sphere heater, horizontal tube heater, and vertical surface heater, the sphere diameter, tube outer diameter, and distance from the leading edge are usually selected as Lx. For wire and flat plate heaters, Laplace reference length, lc, is used. U.S. Natural Bureau of Standards suggested Breen & Westwater correlation could be used to consider the heat transfer of pool film boiling but the used heater geometries were not declared [7]. For this situation, the existing test data on hydrogen film boiling heat transfer are accumulated, and several comparison studies are performed to validate the applicability of these correlations aiming at different heater geometries in hydrogen film boiling events on one hand, on the other hand to confirm the applicative heater geometries of Breen & Westwater correlation. When the diameter of a horizontal tube decreases to a quite small value, a film boiling heat transfer from horizontal wire could be supposed. In other words, there might be no essential difference in film boiling characteristics between at horizontal tube and at horizontal wire. Therefore, the film boiling at horizontal tube and wire heaters are analyzed comprehensively. Fig. 6 displays the comparisons of experimental and calculated heat fluxes, and the calculated results are obtained by Bromley correlation and Breen & Westwater correlation, respectively. It shows within the considered boiling cases, Bromley correlation may underestimate the heat transfer of hydrogen film boiling, even for the case with a horizontal tube heater that Bromley correlation was proposed specially for this geometry condition. Relatively, Breen &

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N2 Vertical h0fg ¼ hfg ð1 þ 0:34cpv DT=hfg Þ2

Lx ¼ L, C1 ¼ 0:943, m ¼ 1=4

Lx ¼ lc , C1 ¼ 0:425, m ¼ 1=4 h0fg ¼ hfg þ 0:4cpv DT

h0fg ¼ hfg ð1 þ 0:34cpv DT=hfg Þ2

Lx ¼ lc , m ¼ 1=4

Westwater correlation gives better predicted results, and most of the data locate within the deviation band of þ10% to 20%. Figs. 7e9 display the comparisons of experimental and calculated heat fluxes for hydrogen film boiling at sphere heater, flat plate heater and vertical heater, respectively, and the representative correlations aiming at the three heater geometries and Breen & Westwater correlation are used to solve the heat fluxes. For the film boiling heat transfer at sphere heaters, Merte & Clark correlation attains a good agreement with the deviation of ±12.5%. Breen & Westwater correlation overestimates the heat transfer of hydrogen film boiling at sphere heater, and the deviation is within 30%. For the heat transfer at flat plate surfaces, both of Berenson and Breen & Westwater correlations give relatively large predicted deviation, and the maximum deviations are approximately 40%. For the film boiling from vertical surfaces, Hsu & Westwater correlation generally underestimates the heat transfer intensity, and the maximum deviation is about 60%. Breen & Westwater correlation obtains a better agreement, and the deviation is within 40%. Based on the above analysis, it may conclude that Breen & Westwater correlation could reveal the heat transfer characteristics of hydrogen film boiling, no matter the heater geometries used. Therefore, Breen & Westwater correlation

!1=4 mv LDT

l3v rv ðrl rv Þgh0fg

[23]

hL ¼ 0:943 Hsu & Westwater,

Berenson, [32]

NuB ¼ 0:425 RaB $

s gðrl rv Þ

hB

Breen & Westwater, [7]

[41]

1=8

h0fg

cpv DT

mv DT l3v rv ðrl rv Þgh0fg 1=2 s ¼ 0:37 þ 0:28 gD2 ðrl rv Þ " !#1=4

!1=4

C1 ¼ 0:37 þ 0:28 Dlc

Wire

Horizontal plate

e

O2, N2 H2, He

N2

N2

Spherical heater Lx ¼ D, m ¼ 1=3, C1 ¼ 0:15 h0fg ¼ hfg þ 0:5cpv DT #1=3 "

h0

NuD ¼ 0:15 Ra$cpvfgDT

Merte & Clark,

cpv DT

NuLx ¼ C1 RaLx $

Lx ¼ D, C1 ¼ 0:62, m ¼ 1=4 h0fg ¼ hfg þ 0:5cpv DT D$DT$Pr

!1=4 l2v rv ðrl rv Þgh0fg cpv

hD ¼ 0:62 Bromley, [24]

Correlations in references Authors

Table 2 e Correlations on pool film boiling heat transfer from different geometry heaters.

h0fg

Unified formation " !#m

Geometry

Horizontal tube

Cryogenic fluids tested


Fig. 6 e Comparison of experimental and calculated heat fluxes for hydrogen film boiling at horizontal tube and wire heaters [39, 16].


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Fig. 7 e Comparison of experimental and calculated heat fluxes for hydrogen film boiling at sphere heaters.

Fig. 8 e Comparison of experimental and calculated heat fluxes for hydrogen film boiling at flat plate heaters.

could be recommended to guide the preliminary design of hydrogen heat transfer systems.

hydrogen for all of the three cases. Knowing MHF, the corresponding wall superheat can be solved based on Breen and Westwater correlation. It also exhibits that an overestimation of DTl could be resolved by Eq. (19) with CL ¼ 0.16 for the highpressure case. For this reason, a modification of CL ¼ 0.031 is proposed based on the analysis of experimental data, and the predicted results of the modified correlation are also listed in Table 3. For MHF, the maximum deviation of the modified correlation is approximately 20%, which is more accurate than the previous correlation. Moreover, it shows that a rise tendency of DTl with the increase of system pressure can be seen from the experimental data but an inverse tendency from the prediction results. To assess the pressure effect on DTl of hydrogen, more detailed experimental data should be supplied in the future.

Minimum heat flux The Leidenfrost temperature signifies the critical value between transition and film boiling regimes. Zuber utilized Taylor instability theory by considering the critical surface wavelength that creates surface waves of sufficient amplitude to break down the vapor film and carries liquid to the heat surface and proposed the following correlation for predicting MHF of pool boiling [36]. qL ¼ CL $rv $hfg

" #1=4 gsðrl rv Þ ðrl þ rv Þ2

(19)

For the heat transfer analysis of hydrogen, Eq. (19) with CL ¼ 0.16 was suggested by Brentari et al. [7]. The experimental data presented by Merte [13], which represented the whole boiling curve of hydrogen involving the continuous variation from transition boiling to film boiling regimes, could be used to evaluate the prediction accuracy of Eq. (19), and the comparison study is illustrated in Table 2. It can be seen clearly that Eq. (19) with CL ¼ 0.16 significantly overestimate MHF of

Transition boiling As has been introduced in Section (The boiling curve), the heat transfer in transition boiling regime is unstable, and dry portions and rewet portions occurs at the heater surface alternatively. Until now, there is no specified empirical correlation of heat transfer for this regime. To realize the smoothly transition between film boiling and nucleate boiling, a linear

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To obtain a complete boiling curve of hydrogen, the correlations of Eqs. (1), (10) and (18), which are functions of DT, should be resolved first to get the heat flux curves for liquid natural convection, nucleate boiling and film boiling regimes. ONB is simultaneously attained since it locates at the intersection point of natural convection and nucleate boiling curves. Then CHF is calculated by Eqs. (14) and (15) for saturation LH2, and Eq. (17) involved for subcooled LH2. For the calculations of heat flux and wall superheat at MHF, Eqs. (18) and (19) are taken into account comprehensively. Subsequently, the heat flux at transition boiling regime could be given based on Eq. (20). Fig. 10(a) displays the predicted hydrogen boiling curve under 0.1 MPa and Fig. 10(b) illustrates the pressure effect on hydrogen boiling curve. It shows that for the case of 0.1 MPa, heat fluxes at ONB, CHF and MHF are in the order of 10 W/m2, 105 W/m2 and 103 W/m2, respectively, and the corresponding wall superheats are approximately 0.1 K, 3 K and 4 K. Moreover, it can be seen from Fig. 10(b) that pressure has a significant influence on MHF but a little influence on liquid natural convection and film boiling heat transfers. Within the considered pressure range, the wall superheat at MHF point varies approximately between 4 K and 15 K.

Conculsions

Fig. 9 e Comparison of experimental and calculated heat fluxes for hydrogen film boiling at vertical plate heaters.

interpolation of heat flux between CHF and MHF was applied in Refs. [18,37] and is also suggested in this paper. q ¼ qCHF

DT DTCHF qCHF qL DTL DTCHF

(20)

Correlations summary and predicted boiling curve Based on the above investigations, the correlations for accounting for the heat transfer of hydrogen in different boiling regimes are summarized in Table 4, which could be used to construct the boiling curve for hydrogen.

Table 3 e Comparison of experimental and calculated MHF for hydrogen. P/Pc

Experiment, [13]

Eq. (19) with CL ¼ 0.16

Eq. (19) with CL ¼ 0.031

qL/W m2 DTl/K qL/W m2 DTl/K qL/W m2 DTl/K 0.079 0.13 0.20

2750 3300 3900

10.5 7.5 6.5

12,040 17,251 24,199

10.1 15.4 22.1

2333 3342 4689

4.3 6.5 6.4

In the present paper, the experimental data on heat transfer of hydrogen pool boiling are surveyed and classified, and a series of correlations for hydrogen boiling heat transfer at different boiling regimes are proposed. Several main conclusions are drawn as follows: (1) In the nucleate boiling regime, liquid pressure has a slight influence on the hydrogen boiling heat transfer. The boiling heat flux is approximately a function of DT2.5, and Eq. (10) could be utilized to reveal the relation of boiling heat flux and wall superheat for hydrogen. (2) Although different correlations have been developed to predict the film boiling heat transfer of hydrogen at different geometries heaters, Breen & Westwater correlation seems to be appropriate for predicting the heat transfer characteristics of hydrogen film boiling no matter the geometries of heater used. (3) For the hydrogen boiling under 0.1 MPa condition, heat fluxes at ONB, CHF and Leidenfrost are in the order of 10 W/m2, 105 W/m2 and 103 W/m2, respectively, and the corresponding wall superheats are approximately 0.1 K, 3 K and 4 K. (4) Liquid pressure has a significant influence on MHF while a little effect on the heat fluxes in liquid natural convection and film boiling regimes. CHF of hydrogen increases with the increase of system pressure firstly and then decreased with further pressure increase, and the maximum of CHF locates at about P/ Pc ¼ 0.35. Generally, the present study is beneficial to understand the boiling heat transfer characteristics of hydrogen, and

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Table 4 e Summarized correlations for hydrogen boiling curve. Regimes

Positions

Correlations

Descriptions

C Llxl ðGrPrÞn $DT

Liquid natural convection

q¼

Onset of nucleate boiling

qONB ¼ C Llxl ðGrPrÞn $DTl ¼ 6309DT2:52 ONB

(DTONB ¼ DTl) for saturation liquid (DTl ¼ DTONB þ DTsub) for subcooling liquid

Nucleate boiling

q ¼ a$DTb

a ¼ 6309, b ¼ 2.52 Independent with pressure.

Critical heat flux

Saturation qCHF;sat ¼

Laminar C ¼ 0.79, n ¼ 1/4 flat plate heater C ¼ 0.53, n ¼ 1/4 wire heater Turbulent C ¼ 0.16, n ¼ 1/3

CK $hfg rv gsðrrl 2rv Þ v

1=4

0.08 < P/Pc < 0.95

CK ¼ 0:18 0:14ðP=Pc Þ5:68 Subcooled ! 0:8 cpl DTsub qCHF;sub r qCHF;sat ¼ 1 þ 0:23 r hfg

Transition boiling

DTDTCHF q ¼ qCHF DT ðqCHF qL Þ L DTCHF

"

Minimum heat flux qL ¼ 0:031rv hfg

#1=4

gsðrl rv Þ ðrl þrv Þ2

Linear interpolation between CHF and MHF

Modified Zuber correlation

(continued on next page)

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Table 4 e (continued ) Regimes

Positions

Film boiling

Correlations

Descriptions Breen & Westwater correlation, Applied for sphere, horizontal tube and wire, flat plate surface, and vertical surface heaters.

0:37þ0:28 lDx

q¼

!1=4 DT

lx $mv DT 0 l3 v rv ðrl rv Þghfg

h0fg ¼ ðhfg þ 0:34cpl DTÞ2 =hfg

Acknowledgements This work is supported by National Natural Science Foundation of China (No.51406142), the Research fund of State Key Laboratory of Technologies in Space Cryogenic Propellants (SKLTSCP1407), Project funded by China Postdoctoral Science Foundation (No.2015T81023) and the Project Supported by Natural Science Basic Research Plan in Shannxi Province of China (No. 2015JQ5133). The Fundamental Research Funds for the Central Universities is also appreciated.

Nomenclatures cp D g Ga

heat capacity at constant pressure, J/(kg K) diameter, m gravitational acceleration, m/s2 gr2 l3 Galilei number, Ga ¼ ml2 c

Gr h hfg h0fg

Grashof number, Gr ¼ n2 x heat transfer coefficient, W/(m2 K) latent heat of vaporization, J/kg effective latent heat of vaporization, J/kg 1=2 1=2 Laplace length, lc ¼ rsl g or lc ¼ gðrlsrv Þ

l

lc L Lx Nu P Pc Pr q Ra T Tc DT DTl Fig. 10 e Predicted saturation boiling curve for hydrogen.

the summarized correlations, listed in Table 4, could provide researchers with reliable mathematical tools to guide the preliminary design of hydrogen heat transfer systems.

DTsub

gbDTL3

distance from the leading edge, m characteristic length, m Nusselt number, Nu ¼ hLl x pressure, Pa critical pressure, Pc ¼ 1.2858 MPa for hydrogen [33]. mc Prandtl number, Pr ¼ lp 2 heat flux, W/m Rayleigh number, Ra ¼ Gr$Pr temperature critical temperature, Tc ¼ 32.938 K for hydrogen, [33] wall superheat, K difference between wall temperature and liquid temperature, K liquid subcooling, K

Greek symbols b volume expansion ratio, K1 l thermal conductivity, W/(m K) m dynamic coefficient of viscosity, Pa s n kinematic coefficient of viscosity, m2/s r density, kg/m3 s surface tension, N/m


Subscripts CHF critical heat flux l liquid phase L Leidenfrost ONB onset of nucleate boiling sat saturation sub subcooling v vapor phase w wall

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