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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
Original
  • 220 Downloads

Abstract

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.

Nomenclature

a

Base radius of the droplet (m)

dσ/dT

Surface tension gradient (N/m-K)

(∂T/∂xi)s

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

g

Gravitational acceleration (m/s2)

hi

Interfacial heat transfer coefficient (W/m2K)

hlv

Latent heat of vaporization (J/kg)

K

Thermal conductivity (W/m-K)

L

Height of the droplet (m)

M

Molecular weight of vapor (kg/mol)

Pv

Vapor pressure (Pa)

q

Average heat flux (W/m2)

ql

Local heat flux (W/m2)

Q

Heat transfer through droplet (W)

r

Radius of the droplet (m)

\( \overline{R} \)

Universal gas constant (J/mol-K)

Rcap

Capillary resistance (K/W)

Rcoat

Promoter layer resistance (K/W)

Rcond

Conduction resistance (K/W)

Rconst

Constriction resistance (K/W)

Rconv

Convection resistance (K/W)

Rint

Interfacial resistance (K/W)

Rma

Marangoni resistance (K/W)

Rth

Thermal resistance (K/W)

t

Time (sec)

T

Temperature (K)

Tcap

Temperature near droplet interface (K)

u

Fluid velocity in x-direction (m/s2)

v

Fluid velocity in y-direction (m/s2)

V

Droplet volume (μl)

w

Fluid velocity in z-direction (m/s2)

x,y,z

Cartesian Co-ordinate

Greek symbols

α

Thermal diffusivity (m2/s)

β

Interface location (degree)

ΔT

Degree of sub-cooling (K)

ρ

Density of condensate (kg/m3)

θ

Contact angle (degrees)

Thickness (m)

σ

Surface tension (N/m)

\( \widehat{\sigma} \)

Condensation coefficient

μ

Dynamic viscosity (kg/m-s)

τ

Shear stress (N/m2).

Non-dimensional parameters

Bi

Biot Number, hia/K

Bo

Bond Number, Δρgr2/ σ

Ma

Marangoni Number, (−dσ/dT×ΔTCpLρ)/Kμ

Pr

Prandtl Number, μCp/K

Re

Reynolds Number, ρuL/μ

Subscripts

c

Properties at promoter layer

sat

Properties at saturation condition

w

Properties at condensing surface

Notes

Acknowledgments

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.

References

  1. 1.
    Bhardwaj R, Ten Kortenaar MV, Mudde RF (2013) Influence of condensation surface on solar distillation. Desalination 326:37–45CrossRefGoogle Scholar
  2. 2.
    Liu L, Jacobi AM (2006) The effects of hydrophilicity on water drainage and condensate retention on air-conditioning evaporators. Presented at the International Refrigeration and Air Conditioning ConferenceGoogle Scholar
  3. 3.
    Khawaji AD, Kutubkhanah IK, Wie JM (2008) Advances in seawater desalination technologies. Desalination 221:47–69CrossRefGoogle Scholar
  4. 4.
    Al-Khayat O, Hong JK, Beck DM, Minett AI, Neto C (2017) Patterned polymer coatings increase the efficiency of dew harvesting. ACS Appl Mater Interfaces 9:13676–13684CrossRefGoogle Scholar
  5. 5.
    Schmidt VE, Schurig W, Sellschopp W (1930) Versuche uber die Kondensation von Wasserdampf in Film- und Tropfenform. Forsch Ingenieurwes 1:53–63CrossRefGoogle Scholar
  6. 6.
    Le Fevre EJ, Rose JW (1965) An experimental study of heat transfer by dropwise condensation. Int J Heat Mass Transf 8:1117–1133CrossRefGoogle Scholar
  7. 7.
    Graham C, Griffth P (1973) Drop size distribution and heat transfer in dropwise condensation. Int J Heat Mass Transf 16:337–346CrossRefGoogle Scholar
  8. 8.
    Leipertz A., Koch G (1998) Dropwise condensation of steam on hard coated surfaces. In: XIth Int. Heat Transfer Conf, p 379–384Google Scholar
  9. 9.
    Eucken A (1937) Energie und stoffaustausch an grenzflächen”, Energy and mass transfer at interfaces, 25:209–219Google Scholar
  10. 10.
    Carey VP (2008) Liquid-vapor phase-change phenomena”, second edition Taylor and Francis Group, LLC, New York:45–472Google Scholar
  11. 11.
    Tanasawa I (1991) Advance in condensation heat transfer, Advances in Heat Transfer (ed.: Hartnett JP, Irvine TF, and Cho IY), 21:56–136.Google Scholar
  12. 12.
    Phadnis A, Rykaczewski K (2017) The effect of Marangoni convection on heat transfer during dropwise condensation on hydrophobic and omniphobic surfaces. Int J Heat Mass Transf 115:148–158CrossRefGoogle Scholar
  13. 13.
    Sikarwar BS, Khandekar S, Muralidhar K (2013) Simulation of flow and heat transfer in a liquid drop sliding underneath a hydrophobic surface. Int J Heat Mass Transf 57:786–811CrossRefGoogle Scholar
  14. 14.
    Chavan S, Cha H, Orejon D, Nawaz K, Singla N, Yeung YF, Park D, Kang DH, Chang Y, Takata Y, Miljkovic N (2016) Heat transfer through a condensate droplet on hydrophobic and nanostructured superhydrophobic surfaces. Langmuir 32:7774–7787CrossRefGoogle Scholar
  15. 15.
    Thickett SC, Neto C, Harris AT (2011) Biomimetic surface coatings for atmospheric water capture prepared by Dewetting of polymer films. Adv Mater 23:3718–3722CrossRefGoogle Scholar
  16. 16.
    Varshney P, Mohapatra S, Kumar A (2017) Fabrication of mechanically stable superhydrophobic aluminium surface with excellent self-cleaning and anti-fogging properties. Biomimetics 2:2CrossRefGoogle Scholar
  17. 17.
    Huang Y, Sarkar DK, Grant CX (2010) A one-step process to engineer superhydrophobic copper surfaces. Mater Lett 64:2722–2724CrossRefGoogle Scholar
  18. 18.
    Miljkovic N, Wang EN (2013) Condensation heat transfer on superhydrophobic surfaces. MRS Bull 38:397–406CrossRefGoogle Scholar
  19. 19.
    Enright R, Miljkovic N, Dou N, Nam Y, Wang EN (2013) Condensation on superhydrophobic copper oxide nanostructures. J Heat Transf 135:091304CrossRefGoogle Scholar
  20. 20.
    Chen X, Weibel JA, Garimella SV (2015) Exploiting microscale roughness on hierarchical superhydrophobic copper surfaces for enhanced dropwise condensation. Adv Mater Interfaces 2:1–6Google Scholar
  21. 21.
    Bansal GD, Khandekar S, Muralidhar K (2009) Measurement of heat transfer during drop-wise condensation of water on polyethylene. Nanosc Microsc Therm Eng 13:184–201CrossRefGoogle Scholar
  22. 22.
    Glicksman LR, Hunt AW (1972) Numeical simulation of dropwise condensation. Int J Heat Mass Transf 15:2251–2269CrossRefGoogle Scholar
  23. 23.
    Abu-Orabi M (1998) Modelling of heat transfer in dropwise condensation. Int J Heat Mass Transf 41:81–87CrossRefzbMATHGoogle Scholar
  24. 24.
    Fatica N, Katz DL (1949) Dropwise condensation. Chem Eng Prog 45:661–674Google Scholar
  25. 25.
    Le Fevre EJ, Rose JW (1966) A theory of heat transfer by dropwise condensation. Presented at the Third International Heat Transfer Conference, ChicagoGoogle Scholar
  26. 26.
    Sikarwar BS, Khandekar S, Muralidhar K (2013) Mathematical modelling of dropwise condensation on textures surfaces. Sadhana 36:1135–1171CrossRefzbMATHGoogle Scholar
  27. 27.
    Kim S, Kim KJ (2011) Dropwise condensation modeling suitable for superhydrophobic surfaces. J Heat Transf 133:081502CrossRefGoogle Scholar
  28. 28.
    Rykaczewski K (2012) Microdrolet growth mechanism during water condensation on superhydrophobic surfaces. Langmuir 28:7720–7729CrossRefGoogle Scholar
  29. 29.
    Miljkovic N, Enright R, Wang EN (2013) Modeling and optimization of superhydrophobic condensation. J Heat Transf 135:111004CrossRefGoogle Scholar
  30. 30.
    Kim H, Nam Y (2016) Condensation behaviors and resulting heat transfer performance of nano-engineered copper surfaces. Int J Heat Mass Transf 93:286–292CrossRefGoogle Scholar
  31. 31.
    Liu X, Cheng P (2015) Dropwise condensation theory revisited: part I. droplet nucleation radius. Int J Heat Mass Transf 83:833–841CrossRefGoogle Scholar
  32. 32.
    Talesh B, Hamid R, Saffari H (2017) Mathematical modeling and numerical simulation of dropwise condensation on an inclined circular tube. J Aerosp Technol Manag 9:476–488CrossRefGoogle Scholar
  33. 33.
    Rose JW (1981) Dropwise condensation theory. Int J Heat Mass Transf 24:191–194CrossRefGoogle Scholar
  34. 34.
    Guadarrama-Cetina J, Narhe RD, Beysens DA, Gonzalez-Vinas W (2014) Droplet pattern and condensation gradient around a humidity sink. Phys Rev E Stat Nonlinear Soft Matter Phys 89:012402CrossRefGoogle Scholar
  35. 35.
    Vemuri S, Kim KJ (2006) An experimental and theoretical study on the concept of dropwise condensation. Int J Heat Mass Transf 49:649–657CrossRefGoogle Scholar
  36. 36.
    Hu HW, Tang GH, Niu D (2015) Experimental investigation of condensation heat transfer on hybrid wettability finned tube with large amount of non-condensable gas. Int J Heat Mass Transf 85:513–523CrossRefGoogle Scholar
  37. 37.
    Adhikari S, Nabil M, Rattner AS (2017) Condensation heat transfer in a sessile droplet at varying biot number and contact angle. Int J Heat Mass Transf 115:926–931CrossRefGoogle Scholar
  38. 38.
    Sikarwar BS (2012) Modeling dropwise condesation underneath inclined textured surfaces. Doctor of Philosophy, Department of Mechanical Engineering, Indian Institute of Technology Kanpur, Kanpur, IndiaGoogle Scholar
  39. 39.
    Elsherbini AI, Jacobi AM (2004) Liquid drops on vertical and inclined surface II: an experimental study of drop geomerty. J Colloid Interface Sci 273:566–575CrossRefGoogle Scholar
  40. 40.
    Elsherbini AI, Jacobi AM (2004) Liquid drops on vertical and inclined surface I: an experimental study of drop geomerty. J Colloid Interface Sci 273:556–565CrossRefGoogle Scholar
  41. 41.
    Kim HY, Lee H, Kang BH (2002) Sliding of drops down an inclined solid surface. J Colloid Sci 247:372–382CrossRefGoogle Scholar
  42. 42.
    Valencia JJ, Peter N (2008) ASM handbook, ASM International, Casting, 15:468–481Google Scholar
  43. 43.
    Incropera PF, Dexitt PD (2007) Fundamentals of heat and mass transfer, Fourth ed. John Wiley and Sons Inc., New York, 1996Google Scholar
  44. 44.
    Tsuruta T, Kato Y (1994) Estimation of condensation coefficient by dropwise condensation method (condensation coefficients of ethylene glycol and water), The Japan Society of Mechanical Engineers, 60(570):158–164Google Scholar

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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