Mathematical modelling of dropwise condensation on textured surfaces

Abstract

Vapor-to-liquid phase change in the form of discrete drops on or underneath a substrate is called dropwise condensation. The process is hierarchical in the sense that it occurs over a wide range of length and timescales. As the associated heat transfer coefficient is much higher than the film and mixed mode of condensation, it is of considerable interest in applications. The present study is focused on mathematical modelling of dropwise condensation process at multiple scales. The model includes formation of drops at the atomistic scale, droplet growth, coalescence, instability, slide off and fall-off, followed by fresh nucleation of liquid droplets. The model shows that the largest stable cluster size in the atomic model matches the minimum drop radius estimated from thermodynamic considerations. The minimum drop radius is insensitive to surface texturing and does not provide controllability at larger length and timescales. A closer examination of droplet distribution over the substrate reveals that small drops are locations of high heat transfer rates, which diminishes with increasing drop radius. The largest drop diameter depends on its stability and hence, the interfacial forces at phase boundaries. Therefore, drop instability controls the heat transfer coefficient in dropwise condensation. Enhancement of heat transfer necessitates that these drops grow with time, become unstable and be swept away as quickly as possible. Enhancement may be achieved either by (i) inclining the substrate or (ii) by creating an interfacial force at the three-phase contact line by a wettability gradient over the horizontal substrate, inducing drop motion. Wall heat transfer and shear stress under moving drops have been determined using a CFD model. A simple model of coalescence has been adopted in this work. Simulation studies on the effect of fluid properties, surface inclination and its wettability condition on drop size distribution, cycle time, heat transfer coefficient, and wall shear stress are comprehensively discussed in the present article.

Appendices

Nomenclature

A :

Surface area, m²

C _p :

Specific heat at constant pressure,W/kg-K

d _b :

Base diameter of drop, m

F :

Force, N

h _lv :

Latent heat of vaporization, J/kg

h :

Heat transfer coefficient, W/m²-K

k :

Thermal conductivity of condensate, W/m-K

N :

Number of nucleation sites, per cm²

p :

Pressure, N/m²

q :

Surface heat transfer, W

q′′:

Average heat flux, W/m²

r :

Radius of drop, m

l :

Distance between two nucleation sites, m

T :

Temperature, K

ΔT :

Temperature difference between the saturated vapor and condensing wall, K

Δt :

Time step, s

u, v, w :

Velocity component in x, y and z directions, m/s

U :

Relative velocity between the condensing wall and the drop, m/s

V :

Volume of the drop, m³

Dimensionless quantities

${\rm C}_f ,\overline{{\rm C}}_f$ :

Local and average skin friction coefficient, 2τ _w /ρU²

${\rm Nu,}\ \overline{{\rm N}}{\rm u}$ :

Local and average Nusselt Number, hd _b /k _c

Pr:

Prandtl Number, μC _p /k _c

Re:

Reynolds number, ρUd _b /μ

Greek symbols

α :

Inclination angle from horizontal, radian/degree

δ :

Thickness of promoter layer, mm

μ :

Dynamic viscosity, Pa-s

ν :

Specific volume, m³/kg

ρ :

Density, kg/m³

σ :

Surface tension of liquid, N/m

$\tau_{\textit{w}} ,\overline{\tau }_{\textit{w}}$ :

Local and average wall shear stress, N/m²

θ :

Contact angle, radians or degrees

Δθ :

Difference in advancing and receding angle, radians or degrees

Subscripts

adv :

Advancing

avg :

Average

b :

Base

coat :

Drop promoter coating

cond :

Conduction

const :

Constriction

crit :

Critical

int :

Interface

l :

Liquid

lv :

Liquid–vapor interface

rcd :

Receding

max:

Maximum

min:

Minimum

sat :

Saturation

sl :

Solid–liquid interface

v :

Vapor

w :

Wall