Heat and Mass Transfer

, Volume 55, Issue 5, pp 1371–1385 | Cite as

Modeling of heat transfer through a liquid droplet

  • Vishakha Baghel
  • Basant Singh SikarwarEmail author
  • K. Muralidhar


In dropwise condensation, the released latent heat passes through the static and sliding droplets to the condensing surface at a rate limited by various thermal resistances. In the present work, numerical simulation of heat transfer through a droplet is carried for one under static and sliding condition. 3-D governing equations with appropriate boundary conditions are solved for the surface, promoter layer and droplet included within the computational domain. Simulations are carried out using an in-house CFD solver. The simulation results are validated against the available data and are found in good agreement. The observations of the present work are: (a) heat transfer through the droplet achieves steady state over a timescale of micro-seconds, (b) the heat fluxes of deformed and equivalent spherical-cap droplet are found to be equal, (c) Marangoni convection is significant for Ma ≥ 2204, (d) convection is the dominant mode of heat transfer during drop slide-off (e) constriction resistance is insignificant for a copper surface of thickness ≤ 2 mm, (f) average heat flux increases with increasing contact angle, interfacial heat transfer coefficient, degree of subcooling and Reynolds number; however, it decreases with increasing Prandtl number of the liquid. These results are useful for sensitivity analysis of various thermal resistances in the mathematical modeling of dropwise condensation underneath inclined surfaces.



Base radius of the droplet (m)


Surface tension gradient (N/m-K)


Temperature gradient vector at the condensing surface (K/m)


Gravitational acceleration (m/s2)


Interfacial heat transfer coefficient (W/m2K)


Latent heat of vaporization (J/kg)


Thermal conductivity (W/m-K)


Height of the droplet (m)


Molecular weight of vapor (kg/mol)


Vapor pressure (Pa)


Average heat flux (W/m2)


Local heat flux (W/m2)


Heat transfer through droplet (W)


Radius of the droplet (m)

\( \overline{R} \)

Universal gas constant (J/mol-K)


Capillary resistance (K/W)


Promoter layer resistance (K/W)


Conduction resistance (K/W)


Constriction resistance (K/W)


Convection resistance (K/W)


Interfacial resistance (K/W)


Marangoni resistance (K/W)


Thermal resistance (K/W)


Time (sec)


Temperature (K)


Temperature near droplet interface (K)


Fluid velocity in x-direction (m/s2)


Fluid velocity in y-direction (m/s2)


Droplet volume (μl)


Fluid velocity in z-direction (m/s2)


Cartesian Co-ordinate

Greek symbols


Thermal diffusivity (m2/s)


Interface location (degree)


Degree of sub-cooling (K)


Density of condensate (kg/m3)


Contact angle (degrees)

Thickness (m)


Surface tension (N/m)

\( \widehat{\sigma} \)

Condensation coefficient


Dynamic viscosity (kg/m-s)


Shear stress (N/m2).

Non-dimensional parameters


Biot Number, hia/K


Bond Number, Δρgr2/ σ


Marangoni Number, (−dσ/dT×ΔTCpLρ)/Kμ


Prandtl Number, μCp/K


Reynolds Number, ρuL/μ



Properties at promoter layer


Properties at saturation condition


Properties at condensing surface



The authors acknowledge the financial support from Science and Engineering Research Board (SERB), Department of Science and Technology (DST), Govt. of India (Project No. ECR/2016/000020).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Vishakha Baghel
    • 1
  • Basant Singh Sikarwar
    • 1
    Email author
  • K. Muralidhar
    • 2
  1. 1.Department of Mechanical EngineeringAmity UniversityNoidaIndia
  2. 2.Department of Mechanical EngineeringIndian Institute of Technology KanpurKanpurIndia

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