Abstract

Film condensation is the main condensation mode for cryogenic fluids with extremely low surface tension. Deep understanding of how fluctuating liquid film affects heat transfer is of great significance to the design and optimization of cryogenic condensers. In this study, a numerical model of nitrogen condensation on a vertical plate is established which adopts an accurate condensation mass transfer model, and the simulation results match well with the experimental data. The transient flow behaviors like thickness, temperature field, velocity field of liquid nitrogen film and various heat transfer coefficients are analyzed. Results show that the condensed liquid nitrogen film undergoes a flow pattern evolution from laminar flow to sinusoidal-like wave flow and then solitary wave flow. Compared to water condensation, nitrogen condensation has a more intense fluctuating flow pattern due to the lower latent heat. Large local heat transfer coefficient could be found at both the trough and peak areas of the film waves, while the lowest value occurs at the front of the wave peak. The analysis shows that the liquid film in the laminar flow zone between the two peaks is thinned to below Nusselt's theoretical value, and the convection effect at the peak is enhanced due to a sharp increase in the wave velocity. Finally, the time domain characteristics of the local heat transfer coefficients reveal that the synergistic effect of intermittent thinning of film thickness and enhanced convection in solitary waves improves the heat transfer coefficient up to 30% compared with the Nusselt's theoretical value for laminar flow.
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Introduction

Condensation process plays a crucial role in the field of refrigeration and liquefaction. Attention has been paid to investigate how to improve the efficiency of condensation heat transfer. By forcing steam to condense into a single droplet rather than a thick liquid film, the condensation heat transfer coefficient can be increased by an order of magnitude [1]. Recently, excellent studies [2], [3], [4], [5], [6] have been made on dropwise condensation for low surface tension fluid, such as toluene, R134a, hexane and pentane. The dropwise condensation forms with the addition of a coating on the metal surface to repel the liquid, these coatings include oil, lubricant and initiated chemical vapor deposition. However, the application of these hydrocarbon-oriented coatings to cryogenic fluids working below 120 K is still a challenge. The dropwise condensation of cryogenic fluids with surface tension less than 10 mN/m has not yet been achieved. For example, nitrogen with distinct molecular structure, has extremely low surface tension and small contact angle and mainly forms film condensation. On the other hand, the effectiveness and durability of these coatings in the liquid nitrogen temperature region are unknown and vapor contamination may be introduced. Therefore, it is still arduous to improve the filmwise condensation coefficient of nitrogen, which significantly limits the heat transfer performance in the cryogenic distillation and other nitrogen liquefaction systems [7, 8].

The Nusselt theory [9] is widely accepted on laminar film condensation of pure steam on a vertical surface. It is considered that the liquid film is the main thermal resistance in the condensation process, in which no fluctuation is assumed on the liquid film surface. According to Kelvin-Helmholtz instability theory [10], when the Reynolds number of liquid film is larger than a critical value, the fluctuation of the vapor-liquid interface will gradually amplify along the flow direction, and the wavy laminar flow is the most universal flow pattern in a medium Reynolds number ranger. This kind of flow behavior is intricate involving the coupling of two-phase flow and heat and mass transfer.

Kutateladze [11] put forward the correlation of condensation heat transfer of fluctuating liquid film based on the data of steam condensation, and got the validation [12] for nitrogen. Karapantsios [13, 14] conducted an experimental study on the condensation process of the steam-air mixture, and tested the liquid film thickness and heat transfer data to indicate that the liquid film fluctuation significantly increases the heat transfer rate. Mudawar [15] analyzed the water film thickness and the statistical properties of the solitary waves, which helped reduce the randomness of the data and eliminated the noise caused by the instability. Tihon [16] particularly emphasized the regulation of the wall shear force exerted by the solitary wave. The experiment and simulation for water both proved the sign change of the wall shear stress in the area around the first depression after the solitary wave, indicating the local reversal of the flow. The generation of reflux promotes the mass mixing of fluids, which changes the temperature gradient and leads to higher heat transfer rate.

It is extremely demanding to measure the liquid film thickness and observe the flow characteristics highly associated with the heat transfer efficiency inside the condensation liquid film, which is hard to realize by experimental technique, especially at cryogenic temperatures [17]. On the other hand, computational fluid dynamics (CFD) methods have achieved some success [18], [19], [20], [21], [22], [23] in describing liquid film flow behaviors with high fidelity. Miyara [18] performed CFD simulations for water on solitary waves with low Reynolds numbers. It was found that the recirculation zone in the solitary wave enhances the heat transfer between the liquid film and the wall, and the size of the vortex increases when Reynolds number increases. Wehinger [21] found that the velocity of the substrate water film is almost half that of that of the stable liquid film in the absence of such high instability. Park [20] analyzed the water wavy behavior of laminar wavy film flow in the Reynolds number range of 200–1000 by the VOF and PLIC method. The numerical results, including average film thickness, wave amplitude, frequency and velocity, are in good agreement with the experimental data. Yu [22] obtained water film thickness, velocity gradient and wall shear stress under different Reynolds number by CFX. The effect of gas-liquid interface shear force on the liquid film flow behavior were also revealed. Yang [23] proposed a numerical model to simulate the evolution of water wavy films by using a non-orthogonal coordinate transformation, and elaborated the heat transfer enhancement mechanism of convection and heat conduction acting on fluctuating condensate film respectively.

At present, both theoretical and experimental results mainly obtained by room temperature fluids [24], [25], [26], [27] have greatly helped reveal heat transfer mechanism of wavy laminar flow condensation. However, the physical properties of cryogenic fluids and room temperature fluids are quite different. It can be obtained by Nusselt prediction that the heat transfer coefficient of nitrogen condensation is one order of magnitude lower than that of water under the same condensation temperature difference and operating pressure. Meanwhile, nitrogen condensation is accompanied by larger liquid film thickness and lower thermal conductivity. In addition, Drosos [28] found that lowering the surface tension would affect the initial position of the fluctuation, and fluids with lower Kapitz number, such as nitrogen, are accompanied by more unstable flow. On the other hand, the current CFD simulations are basically directed at the flow and heat transfer process of falling film with condensation mass transfer processes ignored. In our previous research, a visualized experimental platform [12] for cryogenic condensation has been set up, and the results show that the fluctuation of the liquid film is more intense with the increase of the condensation temperature difference, and the experimental average heat transfer coefficient is increased by 10–30% compared with the Nusselt's theoretical value. Therefore, it is necessary to investigate the flow characteristics of the nitrogen liquid film, revealing the mechanism of wave fluctuation on enhancing the heat transfer coefficient of nitrogen condensation. In this study, a numerical model was built with reference to the experimental physical setup and operating conditions. The flow behaviors, such as the film thickness, temperature and velocity distribution of liquid nitrogen film were investigated. Effects of transient wavy fluctuations on the local heat transfer coefficients were analyzed, which will be helpful for making more active use of the wavy flow in cryogenic condensations.
Section snippets
Computational domain

The whole experimental setup, particularly introduced in previous study [12], consists of a condensation test part, a data acquisition part and a visualization part, as shown in Fig. 1(a). By heating the liquid nitrogen in the reboiler, the nitrogen vapor is generated and enters the condensation chamber. The atmospheric liquid nitrogen in the liquid nitrogen reservoir is used as the cold source, and the high pressure vapor of nitrogen in the condensing chamber turns into liquid on the vertical
Development of liquid film flow pattern

In the following results and discussions, the flow pattern with the strongest fluctuation is analyzed based on the condition of Case 5 in Table 1, where the condensation temperature difference is 11.1 K. The evolution of liquid film with time and location along the wall is shown in Fig. 5. Similarly, as described by Uehara [37], the flow pattern can be divided into three stages: laminar flow, sinusoidal-like wave flow and solitary wave flow. Before 0.1 s, condensate begins to accumulate in the
Conclusion

In this study, a numerical model of nitrogen condensation on a vertical plate is established, which adopts an accurate mass transfer model. The results are in good agreement with the experimental results. The following conclusions are drawn from the discussions:

    1

    Compared with previous results of water, the liquid nitrogen film undergoes similar flow pattern evolution from laminar flow to sinusoidal-like wave flow and to solitary wave flow. Finally, the wall is mainly occupied by fully developed