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

Highlights

• An experimental data base for heat transfer of supercritical water is established.

• 8 existing criteria for calculating q_dht are assessed against the data base.

• A new criterion for q_dht is proposed by considering the effect of G, P and d.

Abstract

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 (q_dht) 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 q_dht. However, the effects of mass flux, pressure and tube diameter on q_dht 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.

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 C_p has a maximum value at the pseudo-critical 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], [3], [4], [5], [6], [7], [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 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 (q_dht) 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], [14], [15], [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 (q_dht), when which is exceeded, the deteriorated heat transfer (DHT) occurs and heat transfer decreases. This boundary heat flux (q_dht) 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 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 (q_dht) were closely related under supercritical pressures and the larger the mass flux the higher q_dht.

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 q_dht under supercritical conditions and the three criteria were given by Eqs. (1), (2) and (3), respectively.

$$\text{Vikhrev et al}.\left[21\right]:q_{\mathit{dht}}=0.4·G$$

$$\text{Styrikovich et al}.\left[22\right]:q_{\mathit{dht}}=0.58·G$$

$$\text{Yamagata et al}.\left[23\right]:q_{\mathit{dht}}=0.2·G^{1.2}$$

From Eqs. (1), (2) and (3), we can see that the q_dht 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 q_dht is worth analyzing.

In addition to mass flux, pressure also has a major impact on q_dht. 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 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], [27], [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 q_dht. 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 q_dht, 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 q_dht. 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, 18.54-mm and 24.38-mm upward tubes. He inferred that tubes with smaller diameters could have the higher q_dht than that with larger diameters under the same experimental conditions. The q_dht 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).

$$\mathit{Bu}=\overline{G{r}_b}\text{/}R{e}_{\text{b}}^{\text{2.7}}$$

where $\overline{G{r}_b}=g{d}^3({\rho }_b-\overline{\rho }){\rho }_b/{\mu }_b^2,\overline{\rho }=\frac{1}{T_w-T_b}{\int }_{T_b}^{T_w}\rho dT$.

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,

$$A\text{c}=Q_b/R{e}_b^{1.625}P{r}_b,Q_b=q_{w}d{\beta }_b/{\lambda }_b$$

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 q_dht. Many scholars [21], [22], [23], [34], [35], [36], [37], [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 q_dht were assessed and compared thoroughly. Finally, by taking the effects of mass flux, pressure and tube diameter into consideration comprehensively, a new criterion for q_dht 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.