Experimental study of critical flow of water at supercritical pressure

Abstract

Experimental studies of the critical flow of water were conducted under steady-state conditions with a nozzle 1.41 mm in diameter and 4.35 mm in length, covering the inlet pressure range of 22.1-26.8 MPa and inlet temperature range of 38-474°C. The parametric trend of the flow rate was investigated, and the experimental data were compared with the predictions of the homogeneous equilibrium model, the Bernoulli correlation, and the models used in the reactor safety analysis code RELAP5/MOD3.3. It is concluded that in the near or beyond pseudo-critical region, thermal-dynamic equilibrium is dominant, and at a lower temperature, choking does not occur. The onset of the choking condition is not predicted reasonably by the RELAP5 code.

Keywords

critical flow / supercritical water-cooled reactor(SCWR) / reactor safety / loss of coolant accident(LOCA)

Introduction

The supercritical water-cooled reactor (SCWR) has been selected as a candidate for the R&D of a generation-IV nuclear power plant in recent years due to its high thermal efficiency and considerable system simplification. During a postulated loss of coolant accident (LOCA) of the reactor, the break will be subject to choking or a critical condition with a maximum discharge flow rate, which can no longer be increased as the downstream pressure decreases further. It would dominate the progress of the transient, the coolant inventory in the core, and, thus, the temperatures of the fuel elements. Therefore, the evaluation of the critical flow rate is a major concern due to its importance to the reactor safety analysis.

So far, the critical flow has been investigated extensively worldwide both experimentally and theoretically. These studies were mostly related to subcritical pressure conditions, as summariized by Levy et al. [1] and Saha [2]. However, for supercritical pressure conditions the experimental data were relatively scarce, though some results have been obtained [3,4].

Under choking conditions, drastic depressurization and vaporization occur, characterized by thermal and mechanical non-equilibrium between two phases. This presents a challenge for the prediction of the critical flow rate due to the lack of adequate knowledge of the interfacial exchanges of mass, energy, and momentum. Various physical models of critical flow have been proposed based on different assumptions: e.g., the homogeneous equilibrium model [5] with the assumption of two phases in thermal equilibrium (same temperature) and mechanical equilibrium (same velocity), the Moody model [6] with the assumption of thermal equilibrium and slip ratio between two phases, and the Henry-Fauske model [7] considering thermal non-equilibrium by an empiric parameter. Generally speaking, these models were validated by experimental data with only limited ranges of parameters, but they cannot be used with confidence out of the ranges due to the lack of adequate experimental data, especially for supercritical pressure conditions.

As part of the National Basic Research Program of China on SCWR, a project on critical flow has been initiated. In a previous study, experiments at subcritical pressure conditions were performed at steady-state conditions with a nozzle [8]. In the present study, the experiment was extended to the supercritical pressure region, aiming at collecting experimental data and getting insight into the chocking phenomena at supercritical conditions.

Experimental facility and procedure

The test section, shown in Fig. 1, is a nozzle 1.41 mm in diameter and 4.35 mm in length with an inlet rounded edge of r=1 mm. The experiment was performed at a supercritical water loop constructed at the China Institute of Atomic Energy (CIAE), as shown in Fig. 2. The de-ioned water is supplied by a three-head piston pump with a maximum pressure of 45 MPa and a flow rate of 2400 kg/h. It passes a dumping tank and a preheater with a capacity of 1100 kW DC and flows upward through the nozzle. The discharge is condensed in a condenser, cooled further by heat exchangers, and flows back to the pump. The flow rate through the nozzle can be controlled by a bypass valve. With this system, the experiment can be conducted at stable conditions.

The major measurements include the flow rate with turbine flowmeters, the pressures upstream and downstream of the nozzle with pressure transducers, the inlet and outlet water temperatures with Ni-Cr/Ni-Si thermocouples, and the voltage and current across the preheater. These parameters are recorded by a data acquisition system.

Experimental results and discussion

The experimental data have been obtained at steady-state conditions, covering the pressure range of 22.1-26.8 MPa and the inlet temperature range of 38-474°C.

Effect of inlet temperature on flow rate

The experimental results are shown in Fig. 3 by displaying G_M against DT_PC, where G_M is the measured mass flux, and DT_PC is the pseudo-critical temperature T_PC minus the inlet bulk temperature T_b. The pseudo-critical temperature T_PC is evaluated by the following expressions as used in Ref. [3]

T_PC=3.0 p+307.6 for 22.1 MPa<p<24.15 MPa,

T_PC=3.767 p+289.0 for 24.15 MPa<p<31.0 MPa.

As seen, the mass flux increases as DT_PC increases (or the inlet temperature decreases). It is noted that, for the region near the pseudo-critical point (i.e., DT_PC around zero), the mass flux increases more steeply with the temperature.

This trend is similar with that observed in the previous experiments for subcritical pressure conditions, when the pseudo-critical point is referred to as the saturation temperature, the positive DT_PC as the region of x₀<0, and the negative DT_PC as the region of x₀>0 (Fig. 4). It is understandable that, associated with a substantial difference in the flow behavior, at either side of the pseudo-critical point the specific volume of water varies significantly.

Comparison of experimental results with homogeneous equilibrium model

The fluid would be at subcritical pressure and two phase conditions at the critical plane. The homogeneous equilibrium model (HEM) is attempted to predict the critical flow rate. The HEM is a basic two-phase critical flow model based on the following assumptions [5]:

1) Both phases have the same velocities.

2) Both phases are in thermal equilibrium condition.

3) The expansion is isentropic.

From the mass and momentum conservation equations of the liquid phase and the vapor phase, the flow rate can be written as

(1)G={\displaystyle \frac{{[\mathrm{2}{(h_{\mathrm{0}}-(\mathrm{1}-x_{\mathrm{e}})h_{\mathrm{le}}-x_{\mathrm{e}}{h}_{\mathrm{ge}})]}^{\mathrm{1}/\mathrm{2}}}^{}}{(\mathrm{1}-x_{\mathrm{e}})v_{\mathrm{le}}+x_{\mathrm{e}}{v}_{\mathrm{ge}}}},

where h is the enthalpy; v the specific volume; x_e the equilibrium quality at the critical plane; the subscripts g and l refer to vapor and liquid, respectively; and 0 refers to the stagnation (inlet) condition.

The critical quality is evaluated by

x_{\mathrm{e}}={\displaystyle \frac{s_{\mathrm{0}}-s_{\mathrm{le}}}{s_{\mathrm{ge}}-s_{\mathrm{le}}}},

where s₀ is the specific entropy at the inlet, and s_le and s_ge are the specific entropies for liquid and vapor at the critical plane, respectively.

The critical condition is defined as

{{\displaystyle \frac{\mathrm{d}G}{\mathrm{d}p}}|}_t=\mathrm{0.}

The mass flux is predicted by Eq. (1) as the downstream pressure decreased successively until a maximum value is attained. Then it is taken as the critical mass flux.

The comparison of the present experimental data with the predictions by HEM is shown in Fig. 5 by depicting G_M/G_HEM versus DT_PC (G_HEM is the predicted mass flux). For the near or beyond pseudo-critical region (DT_PC<0), the deviation of the prediction from the experimental data is generally within 15%. This agreement is reasonable, considering the fact that, in the present calculation, the pressure losses due to the local and friction resistances, which would take a small portion of the total pressure loss, are not accounted. This result suggests that for the inlet condition near or beyond pseudo-critical temperature, thermal equilibrium is a reasonable assumption at the critical plane.

In the previous subcritical pressure experiment, the critical flow exhibits a thermal non-equilibrium feature. This feature is closely related to the inlet pressure and quality as ① for inlet quality greater than a certain value (around 0.1), thermal equilibrium is dominant; and ② in the low quality and low subcooling region, the effect of thermal non-equilibrium is significant at low pressure, tending to decrease as the pressure increases (Fig. 6). Thermal non-equilibrium becomes less important at pressures above 15 MPa. In the present supercritical pressure experiment, thermal equilibrium is dominant for the region near or beyond the pseudo-critical point. This is consistent with the trend observed in the previous experiment.

With the HEM, the critical pressure can be calculated. As exemplified in Fig. 7, with p₀=24 MPa, in the region near the pseudo-critical point, the critical pressure, p_cr, is as high as 15-19 MPa. Higher pressure corresponds to higher thermal conductivity and density of the vapor and smaller bubble or droplet size, associated with more efficient interfacial heat transfer. These would lead to less thermal non-equilibrium compared with low pressure conditions. On the other hand, at higher pressure the mixture density is larger. Thus, during depressurization the expansion is not so strong, and the effect of thermal non-equilibrium on the flow rate is not as significant as that at low pressure conditions.

It is also observed from Fig. 5 that at very large DT_PC, the ratio G_M/G_HEM decreases to as low as 0.65, i.e., the flow rate is over-predicted by up to 50%. This poor prediction by the HEM model indicates that in this region, the flow would not be in a choking condition. This postulation will be discussed further in the next paragraph.

Comparison of experimental results with Bernoulli equation

For an incompressible single-phase flow through an orifice, the discharge coefficient C_D, is 0.61, and the Bernoulli equation is as

(2)G=\mathrm{0.61}\sqrt{\mathrm{2}\rho (p_{\mathrm{0}}-p_{\mathrm{b}})},

where p₀ is the inlet pressure, p_b, the back pressure, and ρ the water density evaluated at the inlet temperature.

The present data are predicted by Eq. (2) with p_b=0.1 MPa, and the comparison between the prediction and the experimental data is shown in Fig. 8. For the region near or beyond the pseudo-critical point, the ratio G_M/G_Bernoulli (G_Bernoulli is the calculated mass flux) is much lower than 1.0, while when DT_PC is larger than about 50°C, the agreement is within 10%.

Since the Bernoulli equation represents the resistance characteristics of a non-choking flow, these results indicate that at larger DT_PC, vaporization and flash would not take place essentially before the liquid leaves the nozzle, and thus the choking would not occur. These results are similar to that from the experiment by Lee and Swinnerton [3], in which at DT_PC of above 100 F (or 38°C), a constant value of C_D=0.6-0.75 is valid for the nozzles of D=1.78-2.5 mm and L/D=0.9-3.5 with a sharp or rounded inlet.

Comparison of experimental results with predictions of RELAP5/MOD3.3

In the reactor safety analysis code RELAP5/MOD3.3, the two-phase choking flow rate is calculated with a model proposed by Trapp and Ransom [9] using a characteristic analysis of a two-fluid model, and the subcooled choking flow rate is calculated with the Burnell model [10]. This code is used to calculate the present experimental data, and the results are shown in Fig. 9. For this calculation, the nozzle is simulated with a “JUNCTION” component, and either the choking or non-choking options are tested.

As seen, the choking model gives a higher G_M/G_RELAP5 ratio (or lower mass flux) than the non-choking model. For the region near or beyond the pseudo-critical point, the agreement is mostly better than 0.7 by the choking model and 0.6 by the non-choking model. As DT_PC increases, the ratio G_M/G_RELAP5 increases. The mass flux ratio approaches 1.0 by the non-choking model, while it exceeds 1.0 and reaches about 1.3 at high DT_PC by the choking model.

For the RELAP5 calculation with the setting non-choking option, the flow will be predicted with the non-choking model regardless of whether the onset of the choking condition is met or not. While with the setting choking option, the flow is predicted with the non-choking model if the onset of the choking condition is not met. In the present experiment, the flow with higher DT_PC exhibits a non-choking behavior, but it is predicted by the code at the choking condition.

Conclusions

Experimental studies of the critical flow of water have been conducted at steady-state under supercritical pressure conditions. The flow characteristic is investigated, and the experimental data are compared with predictions by the homogeneous equilibrium model, the Bernoulli equation, and the RELAP5 code. Under the present experimental conditions, the following conclusions are achieved:

1) The flow rate decreases as the inlet temperature increases, and it varies sharply in the region near the pseudo-critical point.

2) In the region near or beyond the pseudo-critical point, thermal-dynamic equilibrium is dominant, and the critical flow rate can be estimated by the homogeneous equilibrium model.

3) The choking condition does not take place at a low inlet temperature. The onset of the choking condition cannot be predicted reasonably by the RELAP5/MOD3.3 code.

References

[1] Levy S, Abdollahian D, Healzer J, . Critical-flow data review and analysis. <patent>EPRI-NP-2192</patent> 1982

[2] Saha P. A review of two-phase steam-water critical flow models with emphasis on thermal non-equilibrium. <patent>BNL-NUREG-50907</patent> 1978

[3] Lee D H, Swinnerton D. Evaluation of critical flow for supercritical steam-water. <patent>EPRI-NP-3086</patent> 1983

[4] Mignot G, Anderson M, Corradini M. Supercritical fluid critical flow experiment and analysis. In: 3rd International Symposium on SCWR Design and Technology, SCWR-P047, Shanghai, 2007

[5] Starkman E S, Schrock V E, Neusen K F, . Expansion of a very low quality two-phase fluid through a convergent-divergent nozzle. ASME Trans D, J Basic Engineering, 1964, 86(2): 247-256

[6] Moody F J. Maximum flow rate of a single component, two-phase mixture. ASME Trans C, J Heat Transfer, 1965, 87(1): 134-142

[7] Henry R E, Fauske H K. The two-phase critical flow of one-component mixtures in nozzles, orifices, and short tubes. ASME Trans C, J Heat Transfer, 1971, 93(2): 179-187

[8] Chen Yuzhou, Yang Chunsheng, Du Kaiwen, . An experimental study of critical flow of water at steady-state with pressure of up to 22 MPa. In: Proc 7th International Symposium of Heat Transfer, Beijing, 2008

[9] Trapp J A, Ransom V H. A choked-flow calculation criterion for non-homogeneous, non-equilibrium, two-phase flows. Int J Multiphase Flow, 1982, 8(6): 669-681

[10] Burnel J G. Flow of boiling water through nozzles, orifices and pipes. Engineering, 1947, 164: 572-576

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag Berlin Heidelberg