Numerical Investigation of Dropwise Condensation on Smooth Plates with Different Wettability

Condensers are used in various energy-intensive processing industries. Improving their efficiency plays a crucial role in the optimization of energy consumption. Dropwise Condensation is a highly attractive form of heat transfer. This project investigates the effects of different wettability characteristics in dropwise condensation on smooth plates. It involves a Volume of Fluid (VOF) based Computational Fluids Dynamics (CFD) model to carry out the simulations in Open FOAM on smooth plates various wettability’s. Different parameters such as receding angle, advancing angle, and various radii of droplets have been used for investigation. A base case with droplets’ radius of 12.5μm, receding angle, and advancing angles of (34°, 90°) was used. Three sets of simulations were performed. In Set 1, the angles were (34°, 90°). The radius of the first droplet was kept constant at 12.5μm and the radius of the second droplet changes 1:4 progressively. In Set 2, the angles are changed to (154°, 162°), and the radius of the second droplet changes in the same manner above. In Set 3, the angles were kept at (107°, 117°). The radius changes as above. In some cases, the droplets merge and condense. In others, the momentum created by merging of the droplets creates repeated oscillation or even a lift-off, from the cooling plate.


INTRODUCTION:
Condensers are used in various energy intensive processing industries such as refrigeration, power generation and material manufacturing. Their efficiency improvement has a crucial role not only in optimazation of energy consumption, but also in protecting the environment. Enhancement of this equipment requires full knowledge of the condensation process, which is in conjunction with various phenomena such as heat transfer, phase change, and fluid dynamics. Occurrence of different types of condensation process; filmwise and dropwise, depends on existing operational and condensing surface conditions. The work by Mei et al. (2009) in the early 1930's showed that the dropwise condensation is more attractive form of heat transfer (Incropera et al., 2002) due to its much higher surface heat transfer coefficient than the film wise condensation. Advanced condenser designs can reduce device size and pressure drop significantly increasing the energy efficiency. However, such flow patterns cannot be predicted accurately using analytical or simplified models and experimental device design and optimization is often prohibitively costly. Therefore, considerable amount of research efforts has been applied in recent decades to develop computational software tools (e.g., CFD models) in order to simulate and resolve two-phase condensing flows under different operational conditions and geometries. In this project, a VOF based CFD model is used to carry out simulations on a user modified solver of Open FOAM (Openfoam, 2019). For drop wise condensation on smooth plates with various wet ability's.
A series of parametric simulations were carried out varying the wettability of the smooth plate. The effect of different parameters such as contact angle; advancing and receding angles; droplet radius and saturation temperature will be investigated. Dropwise condensation process begins with low nucleation sites, also known as the physical separation of gas, liquid and solid, early stage of phase change due to condensation. Droplets grow by condensation and coalescence with the other droplets till the droplets have grown enough to be removed from the surface by merging and sliding. Once the droplets are removed, the exposed portion of the surface allows to form new droplets and be nucleated (Rose, 1981), thus the dropwise condensation process to continue in a cyclic steady state. Coalescence is the fundamental mode of condensing droplets that are bigger than (Zemach et al., 1992), the smallest thermodynamically viable droplet (e.g., Radius r >2um for water at 1atm) but no previous studies has investigated the direct effect of different wettability characteristics on the phenomena. During the process of the coalscence, the droplets may rumble on the surface and continue to condense. Additionally, in some other cases, due to the momentum created by the merging of the droplets, they may oscillate repeatedly and clear off the surface area. The two events occur depending on the characteristics of the surface. Most influential process that occurs in a dropwise coalscence is condensation. Condensation will allow small amounts of water resulted through reduction of temperature between the main surface and its ambient surface temperature (Chu et al., 2012). Change from its surface heat transfer efficiency has become a key aspect resulted through the dropwise condensation occurred through surfaces, natural or artificial. Coalescence results in two different forms of masses combining together, in this case droplets of water. Which is resulted between two pressures from the droplets from its initial stage and pressure from its final destination, in this case a wet surface. Dropwise calescence is proved to be a common factor for heat exchange between two surfaces that has a temperature change in between the two. There are several characteristics and energy transfers that occur throughout the dropwise coalescence-kinetic energy to gravitational potential energy results from the movement of the droplet to the differed surface. Larger drops that coalescence results from its neighbouring molecules that combine together and tend to become larger in size. Gradually the droplet will reach its maximum weight that it could handle itself and become unstable allowing the larger molecule to break off the initial surface ground on its secondary surface resulting in a heat exchange from kinetic to thermal exchange (Jalil and Karami, 2020;Chu et al., 2012).
The coalescence of droplets in dropwise condensation has been a topic studied and researched by many in the recent past. While contact angle hysteresis and wettability of surface on the coalescence of droplets have been studied by Chu et al. (2012) and Somwanshi et al. (2014), a framework has also been further developed on the basis of this study as well as the disappearance of the interface internally that takes place gradually (Sprittles and Shikhmurzaev, 2012;Enright et al., 2014;Chen et al., 2016) have used the numerical framework for the droplets to jump on adhesion surfaces that are low, induced by model coalescence. Further, there have also been experiments conducted on surfaces that are resistant to water with certain roughness to study the same (Mei et al., 2013). Such scholarship has helped improve the understanding of droplet coalescence of hydrodynamics; however, the studies on the transfer of heat takes place during such coalescence have not been as much. Simulations have been carried out numerically however, wherein the single droplets are studied in a Lagrangian sense (Burnside et al., 1999). The investigations carried out so far have been responsible in expressing droplet coalescence as an immediate event. It is safe to say that these models consist of two droplets merging to become one large droplet and the volume is ideally conserved with the centre at the mean of the weight of the volume of the two previous centres. The heat transfer that takes place at the next stage of the process contains the newly formed droplet to recover from temperature due to collusion as well as the heat transfer rate. This however, does not take into account the unsteadiness of the hydrodynamics in coalescence. The time would vary in reality for each droplet during coalescence, depending on the shape of its base. Droplets that oscillate are also known to affect the area around them. There are thus many such factors that are unaccounted for, that impact the overall rate of heat transfer. Response time and spatial and temporal scales at such initial stages of coalescence process may prove it carry out experiment. For example, the present work uses drop radius varying from (12.5mm ≤ r ≤ 100mm) and time scale of (oms ≤ t ≤ 500ms). Observing these stages can prove to be extremely challenging; hence there are not many studies to validate the effect of heat transfer during condensation.
1.1 Project Management -Appendix 7.2 shows the initial Gantt chart that was designed to follow up tasks throughout the year of final year study. There were many moments that made the project very interesting and enthusiastic to work towards the objectives. However, many setbacks were found upon throughout the year. With the guidance of the supervisor, the setbacks were tackled, and the project was set back on track. During the first few weeks the tasks were complete and everything was on track. However, due to the rush into the holidays and the time constraint that was available during the Easter break, the second simulation was not able to be done on time (shown in red on Gantt chart, Appendix 7.2). Most tasks were finished on time but the delay on the second simulation created a major problem towards the end of the academic year even though the project was due to finish at the start of April, the second simulation resulted in a 3-week set-back. However, the issue was resolved and the simulation was com- The phase change is accounted for by the source term ̇ on the right-hand side. The mass is also globally conserved since the mass which is removed from the liquid side of the interface is added on the vapour side. Conservation of momentum is given as follows: Equation 2.
The effect of surface tension and gravity are respectively accounted for on the momentum source terms at the right side of the equation. The surface tension is modelled according to the continuum method (Deshpande et al., 2012). Conservation of energy is given as follows: Where bulk heat capacity is represented by term , (T) temperature, and bulk thermal conductivity by term λ. The volume fraction α is transferred by the flow field in the equation given below. The sharpening of the interface is vital when simulating two phase flows of fluids. This is artificially obtained in OpenFoam through the introduction of extra compression term (V. (α (1-α) ) in the equation (4). Neuman boundary conditions are imposed for the smooth source term field on all boundaries of the domain. This is to ensure that both integral values of the sharp and smooth source fields remain the same, even after the smearing. Further, the width of the source term field is proportional to the square root of the product of the diffusion constant (adjusted to the mesh resolution for the smearing of the source field on several cells) and the artificial time step. Those source terms in all cells without pure liquid or vapour are set to zero, artificially. By doing so, source terms are shifted into the pure vapour and at the vicinity of the interface remain liquid cells. Thus, velocity field which is calculated is used to transport the interface, in the absence of any source terms. The algorithm for transport for volume fraction field and compression of the interface that is associated works in an efficient manner without the source term field interfering. The source term field which remains is scaled individually on the liquid and vapour side through appropriate scaling coefficients. Mass will thus be globally conserved through this and the evaporating/condensing mass flow corresponds to the net mass glow globally through the interface. These scaling coefficients can be obtained through integration of the smooth source term field in each of the phases that are pure and by comparison to the net mass flow. The following equations can explain these factors: Equation 13 and 14 (Georgoulas et al., 2017) The above are useful scaling factors in obtaining the final source term distribution through the following process: Equation 15 (Georgoulas et al., 2017).
This can be illustrated through the diagram below: The final equation brings about the source term: Equation (16).
The heat capacity is represented by equation (16). While the correction source term responsible for removing artefacts, is explained through the first part; the contribution of the enthalpy is explained through the second part. VOF Solution algorithm used in OpenFOAM summarised in a flowchart in Fig 2.   Fig 2: VOF Solution Algorithm.

Simulation Studies 2.1 Computational Domain & Mesh -
As the coalescence and condensation process of the droplets in the investigation is considered symmetric, a symmetric computational domain was constructed for the investigation. Fig 2 illustrates the representative domain with labels. The constructed computational domain is rectangular cuboids with length, width and height varying in each case. Here, H = W = (2 x r 1 + Rt x r 1 ) and L = (6x r 1 + 2 x Rt x r 1 ), r 1 is radius of the smaller droplet, and Rt ≥2 and changes progressively; is the ratio of the droplets, these values can be found on Table 2 in section 3.3 numerical simulation details below. Computational mesh consists of 144400 hexahedral cells, for the base case.    Table 1: Shows boundary conditions for the simulation domain for the present investigation (Initial Boundary conditions).

Numerical simulation details
In order to investigate the effect of surface wettability, number of simulations (series A) with different parameters such as receding angle, advancing angle and various radii of droplets have been used as shown in Table 2.

S/d-bubble separation, A/a-Advancing angle, R/a-Receding angle, Rt-Ratio
In the present work simulations, a radius of 12.5μm for the first droplet kept constant throughout the investigation. Three sets of simulations were performed altering the wettability by changing advancing and receding angles. In Set 1, the angles were (34°, 90°). The ratio (Rt) is given on Table 2, Rt is the ratio of which the second droplet's radius increases in each case. The angles were (Antonini et al., 2013) changed to (154°, 162°) which has been used and validated in previous work done regarding the droplet receding angles and the radius changes as above. In Set 3, the angles were (Antonini et al., 2013) kept at (107°, 117°). The radius changes as above. The time of the simulations were limited to 500ms in order to observe the initial phase of dropwise condensation 3. RESULTS AND DISCUSSION:

Effect of different surface wettability and different droplet ratio-Series A
In the current section of the present paper, the effect of surface wettability characteristics on droplet condensation is investigated numerically. Simulation series A, three sets of simulations with nine cases in total were carried out varying parameters explained in Table 2. The results of the simulations were as follows. In summary, set 1, Case 1-3 simulations were carried out with the same contact angles, which mean the surface wettability was kept constant. A receding angle of 34° and an advancing angle of 90° creating a hydrophilic condition on the surface. However, the second droplet size was altered with regard to values Rt in Table 2.   Results show these droplets undergo the primary mode of coalescence, where the two droplets merge together with time forming a single droplet with a larger mass and condense gradually. This also shows the effect of various droplet radii. In case 1 and 2 the droplets merger and condense slowly, where as in case 3 the droplets merge and condense at a much faster rate given the same time proving the droplet size effect the condensation rate. Set 2, Case 4-6 were carried out with a different contact angle to that of Set 1 altering the wettability of the condensing surface. A receding angle of 154° and an advancing angle of 162° creating a super hydrophobic condition on the surface. The aspect ratio of the droplets alters as shown in Table 2. Simulation results in these cases show that the two droplets merges, the momentum created by the droplet coalescence creates an oscillation and causes the droplet to lift off from the condensing surface. Once again, the effect of different radii can be observed from the results. In case 1 the droplets undergo the process and lift off within the first 40ms. Cases 5 and 6 undergo the same process, but in much slower time, approx. 100ms and 200ms respectively. This shows that the droplet with larger radius takes longer time to achieve the sufficient momentum in order to overcome the surface tension and lift off.   In this Set 3, Case 7-9, were performed with different contact angles altering the wettability of the condensing surface. A receding angle of 107° and an advancing angle of 117° creating a hydrophobic condition on the surface. The aspect ratio of the droplets alters as in Set 1 and Set 2. Results in these cases shows that they undergo a similar process to case 1-3. However, the momentum created by merging of the two droplets were insufficient to overcome the surface tension and lift off, hence the merged droplets gradually condense with time. Once again, it can be observed that the size of the droplet plays a vital role in the initial phase of condensation as the droplet with larger radius condenses quicker than the two smaller droplets. Simulation series (A) was successfully completed and applicable data was obtained. Results obtained in the present work showed varying wettability have a significant effect on the dynamics of the two droplets after the coalescence points. This should significantly affect the transient condensation rate during the coalescence process; however, this will be further discussed with a more quantitative analysis by comparing the transient condensation rate during the phenomenon (Hasan et al., 2020).

Effect of different surface wettability with different gravity vector orientation-Series B
In this section of the paper, the effect of gravity vector on the droplets coalescing on surfaces with different wettability characteristics is investigated numerically. For this, cases 1-9 from series (A) with altered wettability's of the condense surface and various radii of droplets were utilised, but with an additional gravitational vector. Simulation series B, three sets of simulations with nine cases in total were carried out varying parameters explained in Table 2.
The results of the simulations were as follows;   In summary, Series B, set 1 Case 1b-3b carried out with same parameters as Set 1, Series A. wettability of the surface was set to hydrophilic conditions utilising the values found in Table 2. An additional gravity vector is applied symmetrically in x-direction of the domain. The simulation results obtained are presented by rotating 90deg in order to show the relationships between gravity, surface tension and inertial forces. It can be observed, that case 1b to 3b undergoes a very similar process to case 1-3 in series A. droplets merges together and forms singular droplet with larger mass and condense gradually. However, small differences in values can be found when the values are observed carefully. Fig 17-21 illustrates that these cases 4b to 9b simulations undergoes very similar processes of coalescence as the series A. Case 4b-6b droplets coalescence and take off from the condensing surface by the momentum created. Similarly in cases 7b-9b droplets merge and condense gradually. In summary, as the results found in both series of simulations are similar, this shows that for the considered time scale gravity orientation does not play a major role, whereas the surface tension and inertial forces of the phenomenon are much more important for the considered spatial and temporal scale.

Effect of different wettability characteristics on condensation rates of the phenomenon
In this section of the present paper, the results obtained in simulations series A, are examined in a more quantitative manner by comparing the volume integrals of transient condensation rates against time.

Effect on the volume integral of condensation rate of droplets with different Radii
The ascending gradient shown in Case 1-2 shows that the volume of the droplet increases and stay steady as it undergoes coalesces process, meaning no condensation in the given time scale. Case 3 however shows slight increase volume, then rapid decrease in the gradient means decrease of volume integral; hence faster condensation rate.    Graphs in Fig 25 shows the two droplets with smaller radii lifts off n clear the surface with the rapidly decreasing gradient reaching 0 Kgm-3s-1, meaning no or very little condensation. Graph for case 6 shows there is some unsteady change in the volume integral as the droplets merge and a change shape due to the momentum created, eventually lifts off and clears the surface.
From Fig 26, it can be obtained that once the droplets merge the volume integral does not change much in the cases 7 & 8. However, graph for case 9 shows how volume integral decreases over time; hence slowly condensing over time.

Effect of the different wettability surfaces on volume integral of condensation rate
In this section of the present paper, results obtained from simulations series (A) analysed in order to understand the effect of different wettability surfaces on the droplet condensation process. Simulations were performed in three sets. Radii of the droplets were kept constant and wettability of the surface is altered by varying the contact angle. Wettability of cases 1, 4 & 7 changes from hydrophilic, superhydrophobic and hydrophobic respectively. Graph shows an increase in gradient for case 1, this means the two droplets merge and form a larger mass but does not go pass the primary mode of condensation. Case 4 shows rapid decrease in gradient and reaches 0 Kgm-3s-1 under 40us, hence the droplets has cleared off the domain.   Fig 27 and 28 shows the results for cases with droplets radii of 12.5um & 50um, 12.5um and 100um for the same wettability parameters discussed in Fig 26.

Effect of different wettability characteristics on condensation rates of the phenomenon with a different gravitational orientation.
In this section of the present paper, the results obtained in simulations series B, are examined in a more quantitative manner by comparing the volume integrals of transient condensation rates against time. Fig (29-34) shows the effect of gravity vector on the droplets coalescing on surfaces with different wettability characteristics. For this, cases 1-9 from series (A) with altered wettability's of the condense surface and various radii of droplets were utilised, but with an additional gravitational vector. As discussed earlier in results found in both series of simulations are similar, showing that for the considered time scale gravity orientation does not play a major role, where-as the surface tension and inertial forces of the phenomenon are much more important for the considered spatial and temporal scale (Uddin et al., 2021).

CONCLUSION:
In this present work, an enhanced VOF based CDF model was used on a user modified solver of Open FOAM software, the solver was used and validated in the previous work done by Georgoulas et al. (Georgoulas et al., 2017). This modified solver was then applied to carry out series of simulations of the process of droplet coalesence in dropwise condensation. Simulations were carried out varying the wettability of the smooth plate to a given time scale. The effect of different parameters such as contact angle; advancing and receding angles, droplet radius and saturation temperature were investigated. Series A and B including a total of 18 cases over range of (12.5mm ≤ r≤ 100mm) and aspect ratio of (2≤Rt≤ 8).
Simulation results then were used to understand the effect of different surface wettability in the initial stages (oms ≤ t ≤ 500ms) of dropwise condensation.
The results were then further analysed in a quantitative manner by comparing the volume integral of condensation rate with time and the effect of droplet dynamics in the initial stages coalescence were investigated. Two main events occurred during the process. In some cases, the droplets merged and continued to condense. In other cases, due to momentum created by merging, droplets oscillated and lifted off from the condensing surface. These two events occurred depending on the characteristics of the surface. Simulations were successfully completed. Results from investigation gave a clear understating of the phenomenon during the initial stages of dropwise condensation for the given time scale as discussed in detail. Results from series A showed a significant effect of the droplets dynamics when surface wettability was varied. Results from simulation series B did now show major difference of the phenomenon when the gravity vector orientation is changed. To conclude, the results from present investigation help understand and adds to the existing knowledge of the fundamentals of coalescence of dropwise condensation.

ACKNOWLEDGEMENT
It gives me immense pleasure to have completed this dissertation and acknowledge the support and guidance that I have received throughout the process. I would like to thank my supervisor, Dr. Anastasios Georgoulas, who has not only guided me at every step of the project but has been extremely encouraging throughout. He has shown more confidence in me than I had myself. His eye for detail and constructive criticism has only pushed me to improve my work. I would also like to thank my fellow course mates, whose inputs have been insightful and whose support has been extremely comforting. In addition to this, I would like to thank my family for giving me the opportunity to pursue my passion and for patiently supporting me always. I hope I have managed to do justice to this project and am hopeful that this will be the beginning of many more projects.

CONFLICTS OF INTEREST:
The authors declared that there is no conflict of interest in this article.