A new criterion for the onset of heat transfer deterioration to supercritical water in vertically-upward smooth tubes

Applied Thermal Engineering 151 (2019) 66–76

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Research Paper

A new criterion for the onset of heat transfer deterioration to supercritical water in vertically-upward smooth tubes

T

Xiangfei Kong, Huixiong Li , Qian Zhang, Kaikai Guo, Qing Luo, Xianliang Lei ⁎

State Key Laboratory of Multiphase Flow in Power Engineering, Xi’an Jiaotong University, Xi'an 710049, PR China

HIGHLIGHTS

experimental data base for heat transfer of supercritical water is established. • An criteria for calculating q are assessed against the data base. • 8A existing • new criterion for q is proposed by considering the effect of G, P and d. dht

dht

ARTICLE INFO

ABSTRACT

Keywords: Heat transfer Deteriorated heat transfer Supercritical water Criterion

The heat transfer characteristics of supercritical water, Deteriorated Heat Transfer (DHT) in particular, is crucial to the safety of supercritical devices. It became imperative to develop a new criterion with extensive applicability and high prediction accuracy, to determine whether the operating conditions are safe. To achieve this, a broad-based review was conducted to collect experimental heat transfer data of supercritical water flowing in vertically-upward smooth tubes. The experimental data base consisted of 9705 data points, and the numbers of the Non-DHT and DHT cases were 109 and 64, respectively. Based on the data base, 8 criteria for predicting critical heat flux causing the onset of the DHT (qdht) were assessed and compared thoroughly. It was observed that Yamagata’s criterion showed the best prediction accuracy among the existing criteria. However, Yamagata’s criterion was still insufficient in predicting the DHT conditions. For improvement, further analyses were conducted and the author believed that increasing pressure and decreasing the tube diameter might suppress the occurrence of the DHT under the same conditions, and lead to an increase in qdht. However, the effects of mass flux, pressure and tube diameter on qdht were not considered simultaneously by the earlier researchers. Thus, this paper recognized the need and developed a new criterion which considered the effects of mass flux, tube diameter, and pressure comprehensively. The prediction accuracy of this new criterion for the Non-DHT and DHT were both higher than 90%, and the overall prediction accuracy was 94.25% which was higher than that of existing criteria.

1. Introduction Supercritical pressure water are characterized by a pressure greater than the critical pressure of 22.064 MPa. As is well known, the line of distinction between liquid and gas disappears under supercritical conditions. Hence, the departure from nucleate boiling (DNB) which occurs regularly under subcritical pressures could be avoided under supercritical conditions. Moreover, the thermal physical properties of supercritical fluids experience a dramatic change in the region near the pseudo-critical point (as shown in Fig. 1) and the heat transfer could be enhanced greatly because the Cp has a maximum value at the pseudocritical point [1]. Because of the superior heat transfer performance,

⁎

supercritical fluids have been widely used in many industrial systems, such as Circulating Fluidized Bed boilers (CFB boilers) [2–8] and Supercritical Cold Water Reactors (SCWRs). Pioro et al. [9] identified that the thermal efficiency of SCWRs could be improved to circa 40% or more against the current 33–35%. Moreover, since there is no phase change under supercritical conditions, the steam generators, steam separators and steam dryers could be eliminated which can decrease the operational and capital costs. However, the DHT might happen under some operating conditions, which would lead to the overheating of the heated surfaces [10,11]. Shitsman et al. [12] carried out an experimental study on the heat transfer characteristics of the supercritical water in an 8-mm upward smooth tube. He found that, with

Corresponding author. E-mail address: [email protected] (H. Li).

https://doi.org/10.1016/j.applthermaleng.2019.01.077 Received 23 August 2018; Received in revised form 9 January 2019; Accepted 23 January 2019 Available online 24 January 2019 1359-4311/ © 2019 Elsevier Ltd. All rights reserved.

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Nomenclature Ac Bu C Cp d g G ¯ Gr h N P Pr q qdht Re T

β λ μ ρ ¯ φ

thermal acceleration parameter, A c= Qb/ Reb1.625 Prb, Qb = qw d b/ b ¯ b /Reb2.7 buoyancy parameter,Bu = Gr constant specific heat at constant pressure (kJ/(kg K)) inner tube diameter (mm) gravitational acceleration (m/s2) mass flux (kg/(m2 s)) ¯ b = gd3 ( b ¯) b /µb2 average Grashof number,Gr enthalpy (kJ/kg) number of experimental conditions (–) pressure (MPa) Prandtl number (–), Pr = μCp/λ heat flux (kW/m2) heat flux causing the onset of the DHT (kW/m2) Reynolds number (–), Re = Gd/μ Temperature (K)

Tw Tb

dT

Subscripts b cr max min pc ref w

at bulk temperature at critical point maximum value minimum value at pseudo-critical point at reference value at wall temperature

Abbreviations DHT deteriorated heat transfer DNB departure from nucleate boiling Non-DHT non deteriorated heat transfer

Greek symbols α

thermal expansion coefficient (1/K) thermal conductivity (kW/(m K)) dynamic viscosity (Pa s) density (kg/m3) 1 average density (kg/m3), ¯ = T T w b accuracy rate (–)

heat transfer coefficient (kW/(m2 K))

P = 23.3 MPa, G = 430 kg/(m2 s), and when heat flux was 220.9 kW/ m2, the wall temperature rose monotonously with fluid enthalpy, while a wall temperature peak, i.e., DHT occurred (the maximum amplitude was 593 °C) when the heat flux was 386 kW/m2 under the same conditions. The above result indicated that there exists a critical heat flux (qdht) during the heat transfer process of supercritical water, which when q exceeds, DHT will occur and may further lead to more higher wall temperatures which would exceed the temperature extremes of the heated surface and the tube will eventually rupture. This endorses the importance to develop a criterion with extensive applicability and high prediction accuracy for the design and safe operation of relevant transfer components. As for the critical heat flux under subcritical pressures, a lot of insightful works have been done by many scholars [7,8,13–16] and a thorough review was summarized by Cheng et al. [17]. Cheng [18] carried out an experimental study about critical heat flux of Feron-12

under subcritical pressures in an 8-mm vertical tube. The results suggested that the larger the mass flux the higher the critical heat flux, as shown in Fig. 2. Under supercritical pressures, there is a boundary heat flux (qdht), when which is exceeded, the deteriorated heat transfer (DHT) occurs and heat transfer decreases. This boundary heat flux (qdht) here is similar to the critical heat flux (CHF) under subcritical pressures. Watts et al. [19] investigated the heat transfer characteristics of the supercritical water in a 25.4-mm tube and suggested that a wall temperature peak was observed when P = 25.0 MPa, q = 440 kW/m2, G = 361 kg/(m2 s). The wall temperature peak disappeared when the G was increased to 615 kg/(m2 s) while the P and q remained unchanged. Gu et al. [20] conducted experimental studies on the heat transfer characteristics of supercritical water in a 10-mm upward tube. The results of Gu et al. [20] showed that the heat transfer coefficient profile had a peak value near the pseudo critical point when P = 23 MPa, q = 700 kW/m2, G = 1000 kg/(m2 s) which meant that the heat

Fig. 1. Variations of thermal physical properties versus enthalpy of water at 22.5 MPa. 67

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Fig. 2. CHF versus exit steam quality for different mass fluxes (P = 2.3 MPa, d = 8 mm) [18].

Fig. 3. Density variations versus enthalpy of the supercritical water at different pressures.

transfer was enhanced. However, there was a valley value of the heat transfer coefficient profile which suggested that the DHT occurred when G = 600 kg/(m2 s). The above results indicated that the mass flux and heat flux causing the onset of the DHT (qdht) were closely related under supercritical pressures and the larger the mass flux the higher qdht. Based on the relation between q and G, Vikhrev et al. [21], Styrikovich et al. [22] and Yamagata et al. [23] (the detailed information of those criteria was given in Table 1) proposed their own criteria for calculating qdht under supercritical conditions and the three criteria were given by Eqs. (1), (2) and (3), respectively.

that the degree of the DHT is suppressed with the increase of pressure. The degree of DHT is usually determined by the value of heat transfer coefficient. The lower valley value of the heat transfer coefficient, the severer DHT is. Zhang et al. [25] investigated the heat transfer characteristics of the supercritical carbon dioxide in an upward tube. He concluded that a temperature peak is observed when P = 7.5 MPa while the temperature peak disappears at P = 10.5 MPa. The results of Zhang et al. [25] also means that increasing pressure could depress the degree of the DHT. Many scholars [19,26–28] have indicated that the buoyancy introduced by the large density difference between the near wall region and the core region is the main reason for the DHT under high q/ G conditions. When the heat flux is relative high, the density of the fluid in the near wall region is much smaller than that in the core region and the fluid in the near wall region is accelerated remarkably. Therefore, the shear stress is reduced and the turbulent diffusivity is suppressed. This will reduce the diffusivity of heat and DHT occurs. As can be seen from Fig. 3, the variation of supercritical water’s density becomes milder with an increase in the pressure, and hence the degree of the buoyancy decreases. Eventually, the degree of the DHT is suppressed with the increase in pressure [25]. The above results indicate that higher the pressure, the higher will be the qdht. Unfortunately, some of existing criteria such as Vikhrev et al. [21], Styrikovich et al. [22] and Yamagata et al. [23] criteria, did not consider the effects of the pressure on qdht, which might be responsible for the low prediction accuracy. With the development of the new criterion, the effect of the pressure must be taken into consideration. Besides the mass flux and pressure, tube diameter could also affect qdht. Ackerman et al. [29] carried out an experimental study on the heat transfer characteristics of the supercritical water in 9.4-mm, 11.94-mm,

Vikhrev et al. [21]: qdht = 0.4·G

(1)

Styrikovich et al. [22]: qdht = 0.58· G

(2)

Yamagata et al. [23]: qdht = 0.2· G1.2

(3)

From Eqs. (1), (2) and (3), we can see that the qdht has a direct relationship with mass flux which is consistent with the experimental results. However, the exponentials of these criteria are different which lead to large differences among the three criteria’s prediction accuracies (see Section 3). Thus, the relationship between G and qdht is worth analyzing. In addition to mass flux, pressure also has a major impact on qdht. Lei et al. [24] conducted an experimental study on the heat transfer characteristics of the supercritical water in an upward tube. The author argued that the maximum value of the wall temperature decreases and moves to higher enthalpy with the increase of pressure when q = 300 kW/m2, G = 600 kg/(m2 s). The above phenomenon indicated

Table 1 Existing criteria for the qdht of supercritical water found in open literatures. Authors

Criteria

Application ranges P (MPa)/G (kg/m2 s)/d (mm)/q (kW/m2)

Vikhrev et al. [21] Styrikovich et al. [22] Yamagata et al. [23]

qdht = 0.4·G qdht = 0.58·G

26.5/not stated/20.4/not stated 24/not stated/22/not stated 22.6–31/not stated/7.5–10/not stated

Gabaraev et al. [38]

qdht = 0.2·G1.2

not stated

qdht = 0.79·G ·(P / Pcr )1.5

Cheng et al. [37]

qdht = 1.354 × 10 3· G·(Cp, pc /

Mokry et al. [36] Li et al. [35]

qdht =

Gerrit et al. [34]

pc )

58.97 + 0.745·G

qdht = d·(0.36·(G /d)

1.1)1.21

qdht = 1.942 × 10 6·G 0.795 ·(30

d)0.339·(Cp, pc /

2.065 pc )

68

22.5–31/500–3600/2.5–20/200–1740 24/200–1500/10/160–900 22.5–31/200–1600/7.5–38.1/90–1160 22.5–30/203–1500/7.5–26.0/166–1200

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heat fluxes from 148 to 1810 kW/m2 and inner tube diameters from 3 to 38 mm. The data base has a wide range of experimental parameters which can provide the foundation for the development of the new criterion with broad applications.

18.54-mm and 24.38-mm upward tubes. He inferred that tubes with smaller diameters could have the higher qdht than that with larger diameters under the same experimental conditions. The qdht increased by 40% when the tube diameters were changed from 24.38 mm to 9.4 mm which showed that the larger the tube diameter the more difficult DHT would happen. The same conclusion can be drawn from experimental studies on the heat transfer characteristics of the supercritical carbon dioxide [30,31]. Shiralkar et al. [30] studied the heat transfer characteristics of the supercritical carbon dioxide in different tubes and suggested that the degree of the DHT is weaker in 3.175-mm diameter than that in the 6.35-mm diameter tube. As we discussed earlier, buoyancy is the main reason for the DHT under mixed convection conditions and the tube diameters have a great impact on the formation and development of the buoyancy. Watts et al. [19] proposed a correlation to determine the degree of the buoyancy which is expressed by Eq. (4).

¯ b /Reb2.7 Bu = Gr

2.2. Classification of the heat transfer data for supercritical water In order to compare experimental data and the results predicted by the existing criteria, it was necessary to divide all the experimental cases in Table 2 into two groups: DHT conditions (dangerous conditions) and Non-DHT conditions (safe conditions). The definition of DHT used most frequently in the literatures was proposed by Koshizuka et al. [58]:

R=

In Eq. (6), αexp is the experimental heat transfer coefficient and α0 is the corresponding heat transfer coefficient calculated by D-B correlation which was expressed by Eq. (7) [59]. The DHT is believed to occur when R < 0.3. Otherwise, the heat transfer model is Non-DHT. However, the method mentioned above might mistakenly identify the NonDHT conditions as DHT conditions. Fig. 4 shows the experimental results of Xu et al. [48]. From Fig. 4(a), we can observe that inner wall temperature increases monotonously with fluid enthalpy and heat transfer coefficient profile has a maximum value near the pseudo-critical point. According to Pioio et al. [9], the experimental conditions shown in Fig. 4(a) were Non-DHT conditions while it were considered to be DHT conditions based on Koshizuka’s method [58] because the ‘R’ was lower than 0.3 near the pseudo critical point (see Fig. 4(b)). Kline et al. [60] evaluated different definitions of the DHT and argued that the most straightforward and reliable method to distinguish DHT is the observation of the emergence of peaks in the wall temperature profiles and hence the same method was adopted in this paper. Fig. 5 shows a typical Non-DHT condition and a DHT condition [21,48]. Based on the above method, all the experimental cases in Table 2 were divided into 65 DHT conditions and 109 Non-DHT conditions.

T

¯ b = gd3 ( b ¯) b /µb2 , ¯ = T T T w dT . where Gr b w b From Eq. (4), we can see that the strength of the buoyancy is proportional to the cube of the tube diameter, indicating that the larger the tube diameter the stronger the strength of buoyancy, and the early occurrence of the DHT under mixed convection conditions. Except for buoyancy, thermal acceleration could be the main reason for DHT under forced convection conditions. This special phenomenon and its explanation can be found in [26,32,33] and no details is repeated here. Jackson et al. [26] ever proposed a correlation which is expressed by Eq. (5) as follows,

A c= Qb/ Reb1.625 Prb, Qb = qw d b/

b

(6)

0

(4) 1

exp

(5)

It can be seen from Eq. (5), that the strength of the thermal acceleration is proportional to the tube diameter, implying that the tube diameter has distinct effect on the DHT. To summarize, increasing the mass flux and pressure while decreasing the tube diameter can increase the qdht. Many scholars [21–23,34–38] proposed different criteria, as shown in Table 1. Unfortunately, most of the existing criteria did not consider the effects of mass flux, pressure and tube diameter simultaneously which made the prediction accuracies of the existing criteria relative low. In addition, some criteria were developed based on their own experimental data whose range of experimental parameters was limited and so does with the existing criteria’s applicable range. In this paper, experimental data about heat transfer characteristics of supercritical water were widely collected to expand the applicable range of the new criterion. Eight criteria for the prediction of qdht were assessed and compared thoroughly. Finally, by taking the effects of mass flux, pressure and tube diameter into consideration comprehensively, a new criterion for qdht under supercritical pressures was proposed and the applicability and prediction accuracy of the new criterion was found to be higher than that of all the existing criteria.

(7)

Nu 0 = 0.023Reb0.8 Prb0.4

Table 2 Supercritical water heat transfer data in vertical upward tubes.

2. Collection and classification of the heat transfer data for supercritical water 2.1. Collection of the heat transfer data for the supercritical water A deep understanding of the existing criteria could provide some reference for developing the new criterion. Hence, detailed comparisons between experimental data and the results predicted by the existing criteria were necessary. In this paper a thorough review was conducted and an experimental data base was established which consisted of 9705 experimental points collected from published papers. Table 2 shows the experimental data sources and the corresponding range of experimental parameters. The experimental parameters cover pressures from 22.5 to 31 MPa, mass fluxes from 200 to 2150 kg/(m2 s), 69

Reference

P (MPa)

G (kg/(m2 s))

q (kW/m2)

d (mm)

Shitsman et al. [12] Swenson et al. [39] Herkenrath et al. [40] Vikrev et al. [21,41] Styrikovich et al. [22] Ackerman et al. [29] Ornatsky et al. [42] Yamagata et al. [23] Lee et al. [43] Polyakov et al. [44] Griem et al. [45] Koshizuka et al. [46] Yoshida et al. [47] Xu et al. [48] Kirillov et al. [49] Mokry et al. [36,50,51] Pan et al. [52] Li et al. [53] Zhao et al. [54] Huang et al. [55] Li et al. [56] Gu et al. [20] Shen et al. [57]

23.3–25.3 23, 31 22.5, 24,25 22.6–26.5 24 22.75–31 25.5 22.6–29.4 24.1 24.5 25 31 24.5 23–30 24–24.9 23.9–24.1 22.5 23–26 23 23–25 23–25 23, 25 24

430–449 2150 7,001,500 400–1400 700 406.9–1220 1500 1120–1260 543, 1627 595 500,1000 540 376, 470 800–1200 200–1500 201–1503 400 459.8–1497.5 600, 1500 631–1233 655–1263 600, 1000 420

220.9–385 789 300–1410 300–1160 348–872 157.7–1261 1810 233–930 252–1101 570 300 473 329, 473 200–600 227–884 148–884 300 192–1326.5 275, 800 420–939 466–1102 700, 1000 270

8 9.42 10 8, 20.4 22 9.42, 18.5 3 7.5, 10 38 8 14 9.4 10, 16 12 10 10 17 7.6 7.6 6 6 7.6, 10 19

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3. Evaluation of the existing criteria It is known that many DHT criteria were proposed earlier in history [21–23,34–38] (Table 1). This section deals with the detailed assessment and analysis of the existing criteria that were conducted using the data base (Table 2) which could provide some reference for developing the new criterion. In order to make a quantitative analysis of the prediction accuracy of each DHT criterion, three parameters were introduced, i.e. φNon-DHT (prediction accuracy for Non-DHT conditions), φDHT (prediction accuracy for DHT conditions), φ(overall prediction accuracy) and Eqs. (8), (9), (10) gave the definitions of these three parameters: Non - DHT

DHT

=

=

=

NNon - DHT NNon - DHT

(8)

NDHT NDHT

(9)

' N + NDHT N = Non - DHT N NNon - DHT + NDHT

(10)

where NNon-DHT, NDHT, and N are numbers of Non-DHT conditions, DHT conditions and all experimental conditions in Table 2, respectively. The superscript “ ′ ” means the experimental conditions were predicted correctly by the criteria. As we can see from Table 3, Vikhrev et al. [21] and Styrikovich et al. [22] criteria could predict all the DHT conditions, i.e., φDHT = 100% while φNon-DHT predicted by the two criteria were only 22.93% and 52.29%, respectively. The result shows that the two criteria tended to be on the conservative side. Fig. 6 illustrates the comparison of the experimental values with the calculated values predicted by the Vikhrev et al. [21] and Styrikovich et al. [22] criteria. The two planes in Fig. 6 were drawn according to Vikhrev et al. [21] and Styrikovich et al. [22] criteria and the positive direction of the Z axis were selected as the upward direction of the planes. The red points and blue points represent the DHT conditions and Non-DHT conditions, respectively. The diameter of the points represent the diameters of the experimental tubes. An ideal criterion could distinguish all the Non-DHT and DHT conditions, i.e., all the DHT conditions (red points) were at the top side of the plane while all the Non-DHT conditions (blue points) located on the lower side of the plane. As can be seen from Fig. 6, almost all the red points (DHT conditions) were above the planes which meant that Vikhrev et al. [21] and Styrikovich et al. [22] criteria could predict the DHT conditions correctly. However, it can also be seen that there were some blue points (Non-DHT conditions) above the plane which indicated that Vikhrev et al. [21] and Styrikovich et al. [22] criteria mistakenly identified the Non-DHT conditions as DHT conditions. Hence, Vikhrev et al. [21] and

Fig. 4. Experimental results of Xu et al. [48].

Table 3 Existing criteria evaluated against the data base (Table 2). Criteria

Vikhrev et al. [21] Styrikovich et al. [22] Yamagata et al. [23] Cheng et al. [37] Mokry et al. [36] Gabaraev et al. [38] Li et al. [35] Gerrit et al. [34]

Fig. 5. Typical Non-DHT conditions and DHT conditions [21,48].

70

Non DHT Conditions (109 Cases)

DHT Conditions (65 Cases)

φ

' NNon - DHT

φNon-DHT

NDHT

φDHT

25

22.93%

65

100%

51.72%

57

52.29%

65

100%

70.11%

100

91.74%

56

86.15%

89.65%

103 80 109

94.49% 73.39% 100%

52 63 32

80% 96.92% 49.23%

89.08% 82.18% 81.03%

94 78

86.23% 71.55%

61 51

93.84% 78.46%

89.08% 74.14%

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the variation of the pressure when the heat flux was around the qdht. However, the effect of pressure was not involved in the Li et al. [35] criterion which would have been responsible for the low φNon-DHT. Besides, what we need to understand is that the Vikhrev et al. [21], Styrikovich et al. [22], Yamagata et al. [23] and Mokry et al. [50] criteria were based on their own experimental data. Due to the limitation of the range of the experimental parameters, the above four criteria could not get high prediction accuracy over a wide range of operating conditions. According to the analysis above, the reasons for the low prediction accuracy of the existing criteria can be summarized as: (1) The existing criteria fail to consider the effect of mass flux, pressure and tube diameter on qdht simultaneously. Only taking one or two parameters into consideration is not sufficient for a precise prediction. (2) The criteria may be developed based on limited data sets. (3) Different criteria might have been based on different definitions of the DHT. 4. Development of the new criterion As discussed in the previous section, the mass flux, pressure and tube diameter have a great impact on qdht. However, only a few criteria take the effect of mass flux, pressure and tube diameter on qdht into consideration simultaneously. In addition, the reason why some criteria (such as the Mokry et al. [50] criterion) could not get high prediction accuracy over a wide range of operating conditions is that the range of experimental parameters is limited in the development process of those criteria. Taking into account all those problems, a new criterion was developed based on the data base (Table 2) and all the effects of mass flux, pressure and tube diameter were taken into consideration. As it can be observed from Table 1, the relationship between qdht and G was described by the power function (Eq. (11)). Eq. (12) could be obtained by rearranging Eq. (11).

Fig. 6. Comparison of the experimental values with those predicted by the Vikhrev et al. [21] and Styrikovich et al. [22] criteria.

Styrikovich et al. [22] criteria were inefficient in predicting the NonDHT conditions. The criterion proposed by Mokry et al. [50] was similar to Vikhrev et al. [21] and Styrikovich et al. [22] in terms of the prediction accuracy (φDHT = 96.92%, φNon-DHT = 73.39%). Fig. 7 showed the comparison of the experimental values with the calculated values predicted by the Mokry et al. [50] criterion. From Fig. 7, we can see that a number of blue points (Non-DHT conditions) were above the plane which meant that the criterion proposed by Mokry et al. [50] mistakenly identified the Non-DHT conditions as DHT conditions. As observed earlier, the pressure and tube diameter have a great impact on qdht. However, when we analyzed the forms of Vikhrev et al. [21], Styrikovich et al. [22] and Mokry et al. [50] criteria, we realized that only the effect of mass flux is involved in the above three criteria, which might be the reason for the low prediction accuracies of the three criteria. Unlike Vikhrev et al. [21], Styrikovich et al. [22] and Mokry et al. [50] criteria, the criterion proposed by Gabaraev et al. [23] could predict all the Non-DHT conditions correctly while the φDHT was only 49.23% (as shown in Table 3 and Fig. 8). In practice, the prediction results of the Gabaraev et al. [23] criterion tended to be dangerous because the criterion wrongly identified the DHT conditions (dangerous conditions) as safe conditions (Non-DHT conditions). The overall prediction accuracy of Cheng et al. [37] criterion was relatively high amongst all the existing criteria, but still incapable of predicting the DHT conditions correctly (φDHT was 80%). Besides, as it can be seen from Table 1, the corrected item Cp,pc/βpc was involved in Cheng et al. [37] criterion but is inconvenient for engineering practices as Cp,pc and βpc should be calculated first according to the operating pressures. And same is the case with the Gerrit et al. [34] criterion. Mainly, the performance of the Yamagata et al. [23] and Li et al. [35] criteria were found to be better than the other criteria. However, Yamagata et al. [23] criterion was lacking in predicting DHT conditions (φDHT = 86.15%) and Li et al. [35] criterion was insufficient in predicting Non-DHT conditions (φNon-DHT = 86.23%). As can be seen from the form of Yamagata et al. [23] criterion, only the effect of mass flux on qdht was taken into consideration, while the effect of pressure and tube diameter were not involved in the criterion which was one of the main reasons for the low φDHT. As pointed out by Li et al. [35], the heat transfer characteristic of the supercritical water was very sensitive to

qdht = C1· GC2

qdht C1· GC2

1 = Constant

(11) (12)

From Eq. (12), we can see that the left side of Eq. (12) was a

Fig. 7. Comparison of the experimental values with those predicted by the Mokry et al. [36] criterion. 71

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the common tube diameter in supercritical boilers’ water walls was adopted as the dref. According to Liao [61], the common diameters of water wall tubes were concentrated nearby 20 mm and hence 20 mm was selected as the reference tube diameter. Similarly, Pref was the reference pressure which was the critical pressure of water i.e., Pref = 22.064 MPa ≈ 22.1 MPa. Eq. (13) changes to:

qdht C1· GC2

- 1 = C3

d 20

C4

P 22.1

C5

(14)

After little rearrangement, the final form of the criterion was obtained:

qdht = C1· GC 2 1 + C3

Fig. 9. Experimental results of Shitsman et al. [12].

combination of qdht and G while the right side was a constant. As mentioned above, the pressure and the tube diameter have a great

( ) d dref

C4

and

( ) P Pref

C5

were

introduced into the right side of Eq. (12) to reflect the effect of the pressure and the tube diameter on qdht, respectively.

qdht

1 = C3

C1· GC2 In (dref effect when

( ) d dref

d dref

C4

P Pref

C5

(13)

Eq. (13), dref is the reference tube diameter. The correction term d)C4 was adopted in the Gerrit et al. [34] criterion to reflect the of tube diameter on qdht and the qdht might have been negative d ≥ dref. To avoid this from happening, the correction term

C4

C4

P 22.1

C5

(15)

In order to obtain constants C1, C2, C3, C4, C5 in Eq. (15), multivariate nonlinear regression method was employed in this paper and the premise to use multivariate nonlinear regression method are the multiple sets of specific data – (qdht, P, G, D). Gerrit et al. [34] stated that the qdht must be obtained by variation of the heat flux in sufficiently small steps until DHT was observed. However, the data satisfying the above requirements were extremely rare in the open literature. In the development process of some criteria, such as the Gerrit et al. [34] criterion, the qdht used for multivariate nonlinear regression was higher than the real qdht and this is one of the main reasons why the Gerrit et al. [34] criterion could not get high prediction accuracy for DHT conditions. For example, Fig. 9 gives the experimental results of Shitsman et al. [12]. From Fig. 9, we can know that the variation of the wall temperature versus enthalpy was monotonous when the heat flux was 220.9 kW/m2, which indicated that those conditions were NonDHT conditions. When heat flux was increased to 300 kW/m2, an obvious wall temperature peak (510 °C) was observed which suggested that severe DHT occurred. The accurate qdht should between 220.9 kW/ m2 and 300 kW/m2 and the accurate values of qdht were crucial to the accuracy of the criterion. Fig. 10 shows the schematic representation of qdht. In order to explain the significance of searching and determining qdht under certain flowing conditions for the accuracy of the developed criterion, a solid curve and a dashed curve are assumed as reference curves, as shown in Fig. 10. It can be seen that the criterion (the dot dash line in Fig. 10) could predict all the Non-DHT conditions while the DHT conditions could not be distinguished i.e., φNon-DHT > φDHT if the selected qdht was relative high. As a consequence, the criterion based on the higher qdht would wrongly identify the DHT conditions (dangerous conditions) as Non-DHT conditions (safe conditions) which is unfavorable for the engineering applications. On the contrary, had the selected qdht been relative low, the criterion (the solid line in Fig. 10) could have predicted all the DHT conditions while the Non-DHT conditions could not have been distinguished, i.e., φNon-DHT < φDHT. In order to acquire the precise value of qdht, the dichotomy algorithm was adopted by this paper. Initially, 16 groups of experimental conditions (as shown in Table 4) were selected from the data base (Table 2) which would be used for multivariate nonlinear regression and each group was segregated into two experimental cases whose pressure, mass flux and tube diameter were the same except for heat fluxes. qdown was the maximum heat flux when the heat transfer model was Non-DHT and qup was the minimum heat flux when DHT occurred. Both qdown and qup values were observed values in experimental studies. The qdht should between the qdown and qup, i.e., qdown ≤ qdht ≤ qup. The process of multivariate nonlinear regression could not be carried out unless the precise value of qdht was known. Hence, an initial value of heat flux causing the onset of DHT— q 0dht was assumed as q 0dht = (qdown + qup)/2. Then the constants C1, C2, C3, C4, C5 could be obtained via multivariate nonlinear regression method and so does the heat flux causing the onset of DHT q1dht,cal (superscript “1” means the results of the first iteration, subscript “cal” means the calculated values). The new criterion obtained from the first iteration was evaluated through the

Fig. 8. Comparison of the experimental values with the calculated values predicted by the Gabaraev et al. [38] criterion.

impact on qdht. Hence, two correction terms

d 20

was introduced to the new criterion. In the Gerrit et al. [34]

criterion, the difference between the minimum and maximum values of the tube diameter in his data base was selected as dref (dref = dmax − dmin). However, the form of criteria would change with the range of the data base, more specifically, dmax and dmin. In order to be more specific, 72

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tube diameters from 3 to 38 mm at pressures from 22.5 to 31 MPa, mass fluxes from 200 to 2150 kg/(m2 s) and heat fluxes from 148 to 1810 kW/m2. 5. Assessment of the new criterion Fig. 13 gives the comparison of the experimental values with the calculated values predicted by the new criterion. For convenience, two different views were given in Fig. 13. From Fig. 13(a) we can observe that only a few blue points were on the upper side of the plane indicating that the new criterion could distinguish almost all the NonDHT conditions. In Fig. 13(b), we observe that only a few red points were under the plane which suggested that the new criterion could get a good accuracy in predicting DHT conditions. Table 5 further gives the quantitative comparison of prediction accuracy between the new criterion and the existing ones. As can be seen from Table 5, φNon-DHT and φDHT of the new criterion were both higher than 90% indicating that the performance of the new criterion in predicting both Non-DHT and DHT conditions was better than that of existing criteria. And the overall prediction accuracy was up to 94.25% Supercritical water has been widely used in many advanced industrial systems such as supercritical pressure Circulating Fluidized Bed boilers (CFBs), supercritical water-cooled nuclear reactors (SCWRs) and supercritical solar-thermal power plants, and so forth. In these advanced systems, deteriorated heat transfer (DHT) is undoubtedly one of the major concerns, which may lead to wall temperature peaks and even rupture of the heating pipe. With the help of the new criterion of DHT, the DHT could be hopefully predicted in advance in the design stage of the system, and any related system failure may be avoided. Generally, the new criterion for DHT of supercritical water is of great importance for the design and safe operation of relevant industrial systems.

Fig. 10. The schematic of qdht of the supercritical water. Table 4 Experimental data used for multivariate nonlinear regression. Sources

Shitsman et al. [12] Herkenrath et al. [40] Herkenrath et al. [40] Vikhrev et al. [41] Vikhrev et al. [41] Vikhrev et al. [21] Vikhrev et al. [21] Styrikovich et al. [22] J.Ackerman et al. [29] J.Ackerman et al. [29] Mokry et al. [36,50] Li et al. [53] Li et al. [53] Li et al. [53] Gu et al. [20] Gu et al. [20]

P (MPa)

23.3 22.5 22.5 22.6 22.6 26.5 26.5 24 22.75 24.8 23.9 23 25 26 23 25

G (kg/m2 s)

430 700 1000 400 700 493 1400 700 1220 406.9 1002 459.8 1185 1192.1 1000 1000

d (mm)

8 10 20 8 8 20.4 20.4 22 9.42 18.5 10 7.6 7.6 7.6 10 10

q (kW/m2) qdown

qup

220.9 300 600 300 465 362 930 348 945.7 283.9 681 192 766.7 772.5 700 700

300 500 800 581.5 581.5 454 1160 640 1261 315.5 826 336.2 994.7 991.1 1000 1000

6. Summary 1. The experimental heat transfer data of the supercritical water in vertically-upward tubes were widely collected from the published literatures. The experimental data base consisted of 9705 data points and the numbers of DHT and Non-DHT conditions were 109 and 64 respectively. The data covered pressures from 22.5 to 31 MPa, mass fluxes from 200 to 2150 kg/(m2 s), heat fluxes from 148 to 1810 kW/m2, and inner tube diameters from 3 to 38 mm. 2. A thorough assessment of the existing criteria for qdht of supercritical water was conducted. The research connoted that none of the existing criteria could distinguish the Non-DHT and DHT precisely. Compared to other criteria, Yamagata’s criterion had relatively higher prediction accuracies. 3. Methodically and comprehensively considering the effects of mass flux, tube diameter, and the pressure, a new criterion for the onset of heat transfer deterioration under the supercritical pressure in vertically upward smooth tubes was developed. The prediction accuracies of the new criterion for the DHT and Non-DHT conditions were both higher than 90% and the overall prediction accuracy was 94.25% which was higher than that of the existing criteria. The new criterion is capable of being used for the design and safe operation of ultra-supercritical boilers.

heat transfer data base (Table 2). φNon-DHT and φDHT were calculated separately. If φDHT was far greater than φNon-DHT which means q 0dht was relative low, the real qdht should have been located in the interval B (as shown in Fig. 11). In the second iteration the q1dht = (q 0dht + qup)/2. On the contrary, the real qdht should be located in the interval A (as shown in Fig. 11) if the φNon-DHT was far greater than φDHT. In the second iteration the q1dht = (qdown + q 0dht )/2. qdht was obtained by continuous iterations and the iteration was stopped when φNon-DHT and φNon-DHT no longer changed. Fig. 12 gives the flow chart of the development of the new criterion. After several iterations, the final criterion was given in Eq.(16):

qdht = 0.457· G1.09 1

0.035

d 20

1.96

P 22.1

7.16

(16)

The criteria is valid for water flowing upward in tubes with inner

Fig. 11. The schematic of the dichotomy algorithm for calculating the qdht.

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Fig. 12. The flow chart of developing the new criterion.

Fig. 13. Comparison of the experimental values with those predicted by the new criterion.

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Table 5 Comparison between the existing criteria and new criterion. Criteria

Vikhrev et al. [21] Styrikovich et al. [22] Yamagata et al. [23] Cheng et al. [37] Mokry et al. [36] Gabaraev et al. [38] Li et al. [35] Gerrit et al. [34] New Criterion

Non DHT Conditions (109 Cases)

DHT Conditions (65 Cases)

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φ

N'Non-DHT

φNon-DHT

N'DHT

φDHT

25 57

22.93% 52.29%

65 (100%) 65 (100%)

51.72% 70.11%

100

91.74%

56 (86.15%)

89.65%

103 80 109

94.49% 73.39% 100%

52 (80%) 63 (96.92%) 32 (49.23%)

89.08% 82.18% 81.03%

94 78 104

86.23% 71.55% 95.41%

61 (93.84%) 51 (78.46%) 60 (92.31%)

89.08% 74.14% 94.25%

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