Abstract
The process in which two or more adjoining liquid drops contact each other and merge to form a single drop is referred as coalescence. Drop coalescence is seen in many applications including dropwise condensation of vapor on textured surfaces and in micro-fluidics to enhance scalar mixing with the host medium. Coalescence is initiated with bridge formation at the interface and is followed by large fluid velocities during which the participating liquid media are momentarily set into motion. The origin of coalescence is the internal pressure difference between the initial drops as well as the pressure difference relative to the negative bridge curvature which serves as location of low pressure. The conversion of surface energy to kinetic energy is accompanied by changes in gravitational energy and viscous dissipation. Dissipation here refers to the bulk as well as that occurring at the three-phase contact line over the surface. Contact line motion can be substantial, thus making the surface characteristics central to flow oscillations and decay. After equilibrium is achieved, the single coalesced drop will have a smaller curvature, indicating an irreversible loss of surface energy as dissipation of the coalescence process. The first part of the present chapter examines the literature on the subject and provides a state-of-the-art review. In the second part, an experiment involving two small water drops that are placed adjacent to each other on the hydrophobic surface is discussed. Sessile configuration is considered, and the resulting coalescence process is imaged using a high-speed camera. The three-phase contact line of the combined drop remains unpinned and moves in time, while the liquid bridge relaxes when flow takes place from a region of higher to lower pressure. The digital image sequence is analyzed to find the position of the instantaneous center of mass of the drop, whose movement yields the two velocity components. The possibility of distinct timescales during coalescence is explored from these experiments. The third part of this chapter examines an important application wherein vapor condenses on horizontal and inclined surfaces in the form of drops. Here, drops formed at selected nuclei over the surface grow with time by direct condensation, contact neighboring drops, and grow subsequently by coalescence. At certain instants, the drop volume may be large enough to make them gravitationally unstable, forcing them to leave the surface. While these condensation cycles may last for a few hundred seconds, each coalescence event itself will persist only for a few milliseconds. Coalescence should still be represented in the mathematical model of dropwise condensation, particularly from the viewpoint of local wall shear stresses.
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Notes
- 1.
Authors have studied the pendant configuration also, although for brevity, the primary focus of discussion here is on sessile configuration. The relevant data for pendant configuration are quoted, wherever deemed necessary, for comparative purposes.
- 2.
Coalescence data reported in the previous sections primarily focus on sessile configuration, with qualitative comparison with the pendant configuration [Footnote 1]. The difference between the two configurations, insofar as the estimation of velocity and timescales relevant to the coalescence process, and its bearing on the condensation cycle are concerned, is expected to be small.
Abbreviations
- A :
-
Surface area, m2
- C f :
-
Skin friction coefficient, τw/(½ρU2); overbar indicates time-averaged
- C p :
-
Specific heat of the condensate at constant pressure, J/kg-K
- F :
-
Force acting on the drop, N
- ḡ:
-
Acceleration due to gravity (m/s2)
- h :
-
Heat transfer coefficient, q″/(Tsat − Tw), kW/m2-K
- h lv :
-
Latent heat of vaporization, J/kg
- k :
-
Thermal conductivity of condensate, W/m-K
- m :
-
Mass (kg)
- N :
-
Number of pixels (–) or nucleation site density, cm−2
- M :
-
Number of images (–)
- R :
-
Characteristic length (m)
- t :
-
Time (s)
- U :
-
Characteristic velocity (m/s)
- u c :
-
X-component of centroid velocity (m/s)
- u * c :
-
Non-dimensional x-component of centroid velocity (–)
- νc:
-
Y-component of centroid velocity (m/s)
- ν *c :
-
Non-dimensional y-component of centroid velocity (–)
- x c :
-
X-coordinate of centroid (m); overbar indicates time-average
- x * c :
-
Non-dimensional x-coordinate of centroid (–)
- y c :
-
Y-coordinate of centroid (m); overbar indicates time-average
- y * c :
-
Non-dimensional y-coordinate of centroid (–)
- γ * :
-
Non-dimensional shear rate (–)
- w :
-
Area function (–)
- r b :
-
Base radius of the drop (diameter db), m
- r max :
-
Base radius of the drop at instability due to fall-off, m
- r crit :
-
Base radius of the drop at instability due to slide-off, m
- T sat :
-
Saturation temperature in the vapor phase, K
- T w :
-
Substrate temperature, K
- ΔT:
-
Degree of subcooling, (Tsat − Tw), K
- Δt:
-
Time step, s
- V :
-
Volume of the liquid drop, m3
- α :
-
Inclination angle, radians
- \( \dot{\gamma } \) :
-
Shear rate (s−1)
- μ :
-
Dynamic viscosity (Pa-s)
- ν :
-
Kinematic viscosity (m2/s)
- ρ :
-
Density of fluid (kg/m3)
- σ :
-
Surface tension (N/m)
- τ :
-
Non-dimensional time (–)
- τ w :
-
Wall shear stress, N/m2; overbar indicates time-average
- θ :
-
Contact angle, degrees
- θ adv :
-
Advancing contact angle, degrees
- θ rcd :
-
Receding contact angle, degrees
- θ avg :
-
Average contact angle, degrees
- ∆θ :
-
Contact angle hysteresis, (θadv − θrcd), degrees
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Acknowledgements
Simulations reported in the present study were carried out on High Performance Computing Facility of IIT Kanpur, India.
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Somwanshi, P.M., Muralidhar, K., Khandekar, S. (2018). Coalescence Characteristics of Liquid Drops with Application to Dropwise Condensation. In: Basu, S., Agarwal, A., Mukhopadhyay, A., Patel, C. (eds) Droplet and Spray Transport: Paradigms and Applications. Energy, Environment, and Sustainability. Springer, Singapore. https://doi.org/10.1007/978-981-10-7233-8_7
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