A mathematic model for predicting the volume of water bridge retaining between vertical fin surfacesUn modèle mathématique pour prédire le volume de pont d'eau retenue entre des surfaces à ailettes verticales

https://doi.org/10.1016/j.ijrefrig.2016.04.007Get rights and content

Highlights

  • A mathematical model of water bridge volume was developed for predicting drainage performance.

  • The four-degree polynomial was proposed to describe the meniscus curve of water bridge.

  • The visual experimental setup for observing the front and side shapes of water bridge was built.

  • The contours and volumes of water bridges predicted by the model agree well with the experiment.

Abstract

As the volume of water bridge retaining between vertical fins affects the discharge of water bridge from the fin-and-tube heat exchangers, a mathematical model for predicting the volume of water bridge was developed in this study. In the model, the water bridge volume is calculated by the integral of the meniscus curve along the triple contact line; the triple contact line on the water–fin interface is described by the ellipse function; the meniscus curve on the water–air interface is described by a four-degree polynomial correlated based on the analytical solution of the Young–Laplace equation. The visual experiments were performed to validate the proposed model. It is shown that the predicted shapes of water bridges can qualitatively agree with the visualization images; the predicted volumes of water bridges can decribe 86% of the experimental data within the deviation limit of ±15% and the mean deviation is 10.5%.

Introduction

Fin-and-tube heat exchangers are widely used as evaporators in air-conditioners (Ma et al, 2007, Wang et al, 2000, Zhang et al, 2010), and are always operated under wet conditions (Min, Webb, 2000, Wang et al, 2012, Zhuang et al, 2014). When fin temperature is lower than the dew point temperature, the water vapor in the moist air will condense and form small droplets onto the fin surface (Liu et al, 2010, Ma et al, 2007, Yun et al, 2009). Vertical fins are commonly employed to promote the discharge of droplets from the evaporators by gravity (Korte and Jacobi, 2001). However, the small droplets are still easy to grow up and coalesce with each other, leading to the formation of water bridges between two neighboring vertical fins. The water bridges will block the airflow passage, deteriorating the heat transfer performance and increasing the air pressure drop (Qi, 2013, Yang et al, in press), and so it is meaningful to investigate the drain of water bridges from the vertical fin surfaces.

The key issues of the drain of water bridges cover three aspects, i.e. the contour of contact line (Acero et al, 2005, Chen et al, 2011, Zhang et al, 2006), the contact angles (Akbari et al, 2015, ElSherbini, 2003, Meseguer et al, 1995, Vogel, 1989) and the water bridge volume (Harris, Stocker, 1998, Petkov, Radoev, 2014). The contour of contact line and the contact angles together determine the surface tension acting on the water bridge; the existing investigations indicate that the contact line of the water bridge retaining between vertical fin surfaces takes the shape of ellipse (Zhang et al., 2006), and the contact angles varying along the contact line can be predicted by a third-degree polynomial in terms of azimuthal angle (Akbari et al, 2015, Yang et al, in press). Both the water bridge volume and the gravity force influence the water bridge shape; a model of water bridge volume was developed based on the assumption that the water bridge is hyperboloid (Harris and Stocker, 1998). However, the observation results of water bridge shape obtained by Yang et al. show that the shape of top meniscus is different from that of bottom meniscus for the water bridge retaining between vertical fin surfaces (Yang et al., in press), which means the shape of water bridge should not be described by hyperboloid. For evaluating the drain characteristics of water bridges, a model for calculating the volume of water bridge retaining between vertical fin surfaces is needed.

The model of water bridge volume can be developed by the volume integral of the region enclosed by the water–fin and water–air interfaces. The water–fin interface is a plane surface which is surrounded by the contact line; the water–air interface represents a curved surface which is formed by the rotation of a meniscus curve along the contact line. The key factors to describe the water–fin and water–air interfaces and further to model the water bridge volume are the functions of the meniscus curve and the contact line.

The meniscus curve and the contact line are affected by the forces acting on the water bridge, including the radial gravity and the surface tension (ElSherbini, 2003). The shape of the meniscus curve continually varies with the azimuthal angle and the contact line is elongated along the vertical direction due to these forces. As a result, the effects of the radial gravity and the surface tension should be taken into account in the development of the functions of the contact line and the meniscus curve.

The contact line function was presented in the existing researches. The contact lines of water bridges affected by axial gravity and radial gravity can be described by the circle function (Ahmadlouydarab et al, 2015, Chen et al, 2013, Ferrera et al, 2006, Verges et al, 2001) and the ellipse function (Montanero et al, 2002, Yang et al, in press), respectively. For the “circle” contact line, Chen et al. developed the model of the contact line reflecting the effect of contact angle hysteresis and axial gravity, and found that the contact lines on two solid surfaces may have different equilibrium profiles due to the contact angle hysteresis (Chen et al., 2013); Verges et al. observed the shape of water bridge and investigated the geometry parameters of contact line (Verges et al., 2001). For the “ellipse” contact line, Yang et al. observed the water bridge from both front view and side view, found that the contact line of actual water bridge in the heat exchanger will be elongated due to the effect of radial gravity, and developed an ellipse function for describing the shape of the contact line (Yang et al., in press).

The function of the meniscus curve developed in the existing researches focused on the meniscus curve with fixed shape along the contact line (Marmur, 1993, Sirghi et al, 2006, Vega et al, 2015). Marmur et al. employed the Kelvin equation and contact geometry to calculate the shape of the menisci of water–air interface with the absence of gravity and found that the shape of meniscus curve is always the same for the water bridge under no gravity (Marmur, 1993). Sirghi et al. established the description method of the meniscus curve based on the atomic force microscopy (AFM) pull-off experiments which considered the effect of axial gravity (Sirghi et al., 2006). However, the meniscus curve of the actual water bridge retaining between vertical fin surfaces changes with the azimuthal angle due to the radial gravity, which cannot be described by the models developed in the existing researches.

The purpose of the present study is to present a method for describing the variable meniscus curve under the effects of surface tension and radial gravity, and further to develop a model for calculating the volume of water bridge retaining between vertical fin surfaces based on the description methods of meniscus curve and contact line.

Section snippets

Modeling object and technical road map

Water bridge retaining between two vertical fin surfaces are shown in Fig. 1. The water bridge is enclosed by two types of phase-interfaces, including the water–air interface and the water–fin interface, as shown in Fig. 1(a) and 1(b). The water–air interface is saddle-shaped and is non-axisymmetrical along the radial direction due to the self weight of water bridge, as shown in Fig. 1(c). The shape of the water–fin surface is ellipse, and the direction of major axis of the ellipse is vertical

Design of experiments for model validation

The validation of the proposed model includes two parts: (1) the qualitative validation of the water bridge shape; (2) the quantitive validation of the water bridge volume.

For qualitatively validating the shape of the water bridge, the images of water bridge from both front view and side view should be captured in the present study; for quantitively validating the model, the experimental data of water bridge volume will also be obtained.

During the experimental validation, the experimental data

Visualization results for water bridge shape

Fig. 6 shows the visualization result of the proposed model in predicting the shape of water bridge. The three-dimensional water bridge is regenerated by the commercially available ORIGIN software based on the proposed model, as shown in Fig. 6(a). The comparsions of the front and side views of the water bridge between the predicted results and the experimental photos are shown in Fig. 6(b) and 6(c), respectively. The water bridge size L is 2.5 mm; the fin pitch d is 1 mm; the advancing and

Conclusions

  • (1)

    The volume of water bridge can be calculated by the predicted model which combines the ellipse function and four-degree polynomial.

  • (2)

    The predicted shapes of water bridges can qualitatively agree with the visualization images for both front view and side view.

  • (3)

    The results of volume model agree with 89% of the experimental data within the deviation of ±15% and the mean deviation is 9.4%.

  • (4)

    The minimum contact angle can be approximated to the receding contact angle when the water bridge is close to the

Acknowledgements

The authors gratefully acknowledge the support from the National Natural Science Foundation of China (Grant No. 51576122), the Foundation for Innovative Research Groups of the National Natural Science Foundation of China (Grant No. 51521004) and the Program of Shanghai Academic Research Leader (Grant No. 16XD1401500).

References (31)

  • YunR. et al.

    Air side heat transfer characteristics of plate finned tube heat exchangers with slit fin configuration under wet conditions

    Appl. Therm. Eng

    (2009)
  • ZhangL. et al.

    Performance analysis of a no-frost hybrid air conditioning system with integrated liquid desiccant dehumidification

    Int. J. Refrigeration

    (2010)
  • F.J. Acero et al.

    Liquid bridge equilibrium contours between non-circular supports

    Microgravity Sci. Technol

    (2005)
  • M. Ahmadlouydarab et al.

    Dynamics of viscous liquid bridges inside microchannels subject to external oscillatory flow

    Phys. Rev. E Stat. Nonlin. Soft Matter Phys

    (2015)
  • A. Akbari et al.

    Liquid-bridge breakup in contact-drop dispensing: liquid-bridge stability with a free contact line

    Phys. Rev. E Stat. Nonlin. Soft Matter Phys

    (2015)
  • Cited by (0)

    View full text