Heat transfer during condensing droplet coalescence

Proceedings of the 5th World Congress on Mechanical, Chemical, and Material Engineering, 2019

Dropwise condensation is one of the regimes leading to the best heat performance for managing large heat fluxes. In order to model the transfers in this regime, knowledge of the drop size distribution is fundamental. In this paper, two different approaches are proposed and analyzed to determine this distribution. The first one is based on a statistical model, in which two populations of drops are considered and where the steady state is reached. The second model uses a more direct approach, modeling each drop on the studied surface. Assuming instantaneous coalescence when two droplets overlap, the distribution is calculated in transient regime. The mean value of this distribution in permanent regime is then compared to the distribution deduced from available statistical approach in the literature. Good adequation is obtained for the “big” droplets, while important discrepancies are highlighted for the “small” droplets.

2011

New solutions to the droplet heat conduction equation, assuming that the droplet radius is a linear function of time, are described alongside new solutions to the same equation, assuming that the time evolution of droplet radius R d (t) is known. For sufficiently small time steps the time evolutions of droplet surface temperatures and radii, predicted by both approaches for typical Diesel engine-like conditions, coincide. Both solutions predict lower droplet temperatures and slower evaporation when the effects of the reduction of R d (t) due to evaporation are taken into account. A simplified model for multi-component droplet heating and evaporation, based on an analytical solution to the species diffusion equation, is discussed. This model has been generalised to take into account the effect of coupling between the droplets and the surrounding gas. A new approach to modelling multi-component droplets, including large numbers of components, heating and evaporation is discussed. This new approach is based on the replacement of a large number of actual components with a small number of quasi-components. The evaporation and condensation of n-dodecane (C 12 H 26) are investigated using the molecular dynamics (MD) simulation technique. The values of the evaporation/condensation coefficient are estimated.

2006

Several new approaches to the modelling of liquid droplet heating and evaporation by convection and radiation from the surrounding hot gas are reviewed. The finite thermal conductivity of the liquid, recirculation within droplets, time dependence of gas temperature and the convection heat transfer coefficient are taken into account. The relatively small contribution of thermal radiation to droplet heating allows us to describe it by a simplified model, which does not consider the variation of radiation absorption inside the droplets.

International Journal of Thermal Sciences, 2011

The earlier reported simplified model for multi-component droplet heating and evaporation is generalised to take into account the coupling between droplets and the ambient gas. The effects of interaction between droplets are also considered. The size of the gas volume, where the interaction between droplets and gas needs to be taken into account, is estimated based on the characteristic thermal and mass diffusion scales. The model is applied to the analysis of the experimentally observed heating and evaporation of monodispersed n-decane/3-pentanone mixture droplets at atmospheric pressure. It is pointed out that the effect of coupling leads to noticeably better agreement between the predictions of the model and the experimentally observed average droplet temperatures. In most cases, the observed droplet temperatures lie between the average and central temperatures, predicted by the coupled solution. For the cases reported in this study, the observed time evolution of droplet radii cannot be used for the validation of the model. It is pointed out that the number of terms in the series in the expressions for droplet temperature and species mass fraction can be reduced to just three, with possible errors less than about 0.5%. In this case, the model can be recommended for the implementation into computational fluid dynamics (CFD) codes and used for various engineering applications, including those in internal combustion engines.

International Journal of Heat and Mass Transfer, 2017

Dropwise condensation has been identified as a promising heat transfer mechanism because it can yield heat fluxes up to an order of magnitude higher than typically found in filmwise condensation. Models for dropwise condensation generally assume a statistical distribution of droplet sizes and integrate heat transfer over the droplet size spectrum, considering droplet curvature effects on saturation temperature, conduction thermal resistance, and interfacial resistance. Most earlier studies have assumed a constant heat transfer factor (f = O(1)) to account for the conduction contribution to total thermal resistance. However, f varies with droplet Biot number (Bi) and contact angle (h). Formulations for f with broad ranges of applicability are not currently available. In this study, finite element simulations are performed to determine f and corresponding numerical uncertainties for 0.0001 6 Bi 6 1000 and 10°6 h 6 170°. This spans the active droplet size range considered in most droplet condensation studies (e.g., for water condensing at P atm on a surface 10 K below the ambient temperature, active droplets have 0.0005 < Bi < 300). An explicit correlation is proposed for f and is validated with published results. The proposed correlation can facilitate modeling and analysis of dropwise condensation.

Journal of the Indian Institute of Science, 2019

Introduction Condensation of vapor plays a crucial role in a wide range of large-scale energy systems. In particular, steam power plants and HVAC systems, which, respectively, account for 78% of global electric power generation 1, 2 and 10-20% of total energy consumption in developed countries 3 , rely on the process of vapor condensation. Besides steam power plants and HVAC systems, efficiency of several industrial applications such as water desalination 4-7 , water collection 8-10 and thermal management 11-13 depend on vapor condensation. Therefore, any improvement in the efficiency of vapor condensation process can lead to significant energy savings. Condensation can be categorized as either filmwise condensation (FWC) or dropwise condensation (DWC). Figure 1 shows the schematic and Fig. 2 shows the images illustrating FWC and DWC. In FWC, the condensate forms a liquid film on the surface. This liquid film provides additional thermal resistance to heat transfer between the surface and the vapor. On the other hand, in DWC, vapor forms distinct liquid drops

International Journal of Heat and Mass Transfer, 2010

A simplified model for bi-component droplet heating and evaporation is developed and applied for the analysis of the observed average droplet temperatures in a monodisperse spray. The model takes into account all key processes, which take place during this heating and evaporation, including the distribution of temperature and diffusion of liquid species inside the droplet and the effects of the non-unity activity coefficient (ideal and non-ideal models). The effects of recirculation in the moving droplets on heat and mass diffusion within them are taken into account using the effective thermal conductivity and the effective diffusivity models. The previously obtained analytical solution of the transient heat conduction equation inside droplets is incorporated in the numerical code alongside the original analytical solution of the species diffusion equation inside droplets. The predicted time evolution of the average temperatures is shown to be reasonably close to the measured one, especially in the case of pure acetone and acetone-rich mixture droplets. It is shown that the temperatures predicted by the simplified model and the earlier reported vortex model are reasonably close. Also, the temperatures predicted by the ideal and non-ideal models differ by not more than several degrees. This can justify the application of the simplified model with the activity coefficient equal to 1 for the interpretation of the time evolution of temperatures measured with errors more than several degrees. Crown

Heat Transfer Research, 2008

Nomenclature acoefficient introduced in Eq (9), μ m-b ; bcoefficient introduced in Eq (9); B λ-Plank function, Wμ m-1 m-2 ; B Mmass Spalding number; cspecific heat capacity, Jkg-1 K-1 ; C 1 coefficient in the Plank function, Wμ m 4 m-2 ; C 2coefficients in the Plank function, μ m K; hconvection heat transfer coefficient, Wm-2 K; h 0parameter introduced in Eq (5); kthermal conductivity, Wm-1 K; Kthermal diffusivity, m 2 sec-1 ; Lspecific heat of evaporation, Jkg-1 ; n index of refraction; P-radiation source term, Ksec-1 ; Q aefficiency factor of absorption;

International Journal of Heat and Mass Transfer, 2013

The previously developed kinetic model for droplet heating and evaporation into a high pressure air is generalised to take into account the combined effects of inelastic collisions between molecules in the kinetic region, a non-unity evaporation coefficient and temperature gradient inside droplets. It is pointed out that for the parameters typical for Diesel engine-like conditions, the heat flux in the kinetic region is a linear function of the vapour temperature at the outer boundary of this region, but practically does not depend on vapour density at this boundary for all models, including and not including the effects of inelastic collisions, and including and not including the effects of a non-unity evaporation coefficient. For any given temperature at the outer boundary of the kinetic region the values of the heat flux are shown to decrease with increasing numbers of internal degrees of freedom of the molecules. The rate of this decrease is strong for small numbers of these degrees of freedom but negligible when the number of these degrees exceeds 20. This allows us to restrict the analysis to the first 20 arbitrarily chosen degrees of freedom of n-dodecane molecules when considering the effects of inelastic collisions. The mass flux at this boundary decreases almost linearly with increasing vapour density at the same location for all above-mentioned models. For any given vapour density at the outer boundary of the kinetic region the values of the mass flux are smaller for the model, taking into account the contribution of internal degrees of freedom, than for the model ignoring these degrees of freedom. It is shown that the effects of inelastic collisions lead to stronger increase in the predicted droplet evaporation time in Diesel engine-like conditions relative to the hydrodynamic model, compared with the similar increase predicted by the kinetic model considering only elastic collisions. The effects of a non-unity evaporation coefficient are shown to be noticeable for gas temperatures of 1500 K. The application of the rigorous kinetic model, taking into account the effects of inelastic collisions and a non-unity evaporation coefficient, and the model taking into account the temperature gradient inside droplets, is recommended when accurate predictions of the values of droplet surface temperature and evaporation time in Diesel engine-like conditions are essential.

2013

The instantaneous and time-averaged heat transfer coefficient and wall shear stress in dropwise condensation depend on several parameters such as the physical and chemical texture of the substrate, its inclination, and interfacial properties. These factors affect the shape, size, and drop distribution of the condensing drops. On an inclined surface, contact angle varies over the three-phase contact line and the drops get deformed. The commonly used "two-circle model" approximates drop shape as a part of the sphere with a circular footprint. In the present work, this approximation is relaxed and the shape of the drop is obtained by solving the three-dimensional force equilibrium equation. With the shape prescribed, the critical size of drop at instability is determined. Drop-level flow and heat transfer rates associated with fluid motion within have been determined numerically by solving the 3D Navier-Stokes and energy equations in the true drop geometry with applicable boundary conditions. The dropwise condensation model proceeds from the thermodynamically stable liquid droplets, to growth by direct condensation and coalescence, and drop instability, followed by fresh nucleation. Numerical data obtained from the simulation show that wall shear stress and heat transfer coefficient are sensitive to the prescription of the drop shape. The approach proposed in the present study shows a longer time for the drop to become unstable and a larger critical drop size, both factors lowering the estimates of average wall shear stress and heat flux, in comparison to the two-circle approximation.