Condensers are used in various energy intensive pro-cessing industries such as refrigeration, power gene-ration and material manufacturing. Their efficiency improvement has a crucial role not only in optima-zation 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 dyna-mics. 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 1930s 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 analy-tical 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 com-putational 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 con-densation on smooth plates with various wet abilitys.  

A series of parametric simulations were carried out varying the wettability of the smooth plate. The effect of different parameters such as contact angle; ad-vancing and receding angles; droplet radius and satu-ration 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 re-moved, 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 thermodyna-mically 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 coals-cence, 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 coals-cence is condensation. Condensation will allow small amounts of water resulted through reduction of tem-perature 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. Coal-escence 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 chara-cteristics and energy transfers that occur throughout the dropwise coalescence- kinetic energy to gravi-tational potential energy results from the movement of the droplet to the differed surface. Larger drops that coalescence results from its neighbouring mole-cules 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 Som-wanshi 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 experi-ments 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 hydrodyna-mics; 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 investi-gations 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 drop-let 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 hydro-dynamics 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 super-visor, 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. How-ever, 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-pleted successfully on time with set of results and the project objectives were accomplished; i) Numerical Method, and ii) Governing equations. The numerical method including the governing equations and phase change model are in this section. Both liquid and vapour phases are treated as incompressible, New-tonian fluids. The mass conservation equation is as follows: Equation 1 (Georgoulas et al., 2017).

The phase change is accounted for by the source term ( ρ) ̇ on the right-hand side. The mass is also glo-bally 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: Equation 3 (Antonini et al., 2013).

Where bulk heat capacity is represented by term〖 c〗_p, (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 shar-pening of the interface is vital when simulating two phase flows of fluids. This is artificially obtained in OpenFoam through the introduction of extra com-pression term (V. (α (1-α)〖 U〗_r) in the equation (4).  Equation 4 (Somwanshi et al., 2014),

The artificial compression velocity〖 U〗_r  in equation 4 is calculated using the following equation:  Equation 5 (Georgoulas et al., (2017)

Where the cell surface normal vector represented by term〖 n〗_f, mass flux by f, surface area of a cell by S_f and C_γ  is a coefficient where the value is set between 1 to 4. The relative velocity of the two fluid phases across the interface is represented by term〖 U〗_r. In equation (4), the conservation of the volume fraction α is ensured by the process of compression velo-city〖 U〗_r, the term α (1-α) limits this artificial com-pression approach only in the vicinity of the interface, where 0 < α < 1. The level of compression is depen-dent on the value of  C_γ  (Hoang et al., 2013 and Geor-goulas et al., 2017). VOF model used for this investi-gation in OpenFOAM does not solve equation (4) completely, instead applies a multi-dimensional universal limiter with an explicit solution algorithm in order to achieve a sharp interface and limits the volume fraction between 0 and 1 (Georgoulas et al., 2017). The average over the liquid and vapour phases, weighted with the volume fraction are computed as bulk properties. The following equation explains the process: Equation 6 (Georgoulas et al., 2017),

The VOF method is known to be affected by currents that are non-physical and spurious in the region of the interface. This may be owing to errors in normal vector calculation and interface curvature used. Such errors can be accounted for using the following method: Equation 7 (Georgoulas et al., 2017),

1.2 Phase Change model

The following equation has been used to calculate the evaporating mass flux at the liquid-vapour interface: Equation 8 (Georgoulas et al., 2017). The following equation was used to calculate the interfacial heat resistance: Equation 9 (Georgoulas et al., 2017)

The uncertainty of the parameter γ which lies in the range of 0 and 1 and is considered as the evaporation or condensation coefficient. The amount of liquid evaporated is locally calculated and numerical in-stabilities are avoided by smearing the resulting source term field over a few cells. Equation 9 calcul-ates the evaporating/condensing mass flux and is incorporated in the conservation equations. This can be seen in the equation: Equation 10 (Georgoulas et al., 2017)

The following equation is used for the Net Mass Flow: Equation 11(Georgoulas et al., 2017)

The diffusion equation for the smooth distribution of source terms: Equation 12 (Georgoulas et al., 2017).

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/con-densing mass flow corresponds to the net mass glow 

globally through the interface. These scaling coeffi-cients 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).

Fig 1: The diagram depicts an evaporating bubble, where in the distribution of the final source terms can be viewed in the computational domain (Georgoulas et al., 2017).

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.

2. Simulation Studies

2.1 Computational Domain & Mesh - As the coal-escence and condensation process of the droplets in the investigation is considered symmetric, a sym-metric 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 r1 + Rt x r1 )  and L = (6x r1 + 2 x Rt x r1 ), r1 is radius of the smaller droplet, and Rt ≥2 and changes pro-gressively; is the ratio of the droplets, these values can be found on Table 2 in section 3.3 numerical simulation details below. Computational mesh con-sists of 144400 hexahedral cells, for the base case.

Fig 3 illustrates representative domain for case 1 (Georgoulas et al., 2017). The domain only includes half of the droplets and their surroundings, as two droplets are symmetric along one centre plane. The face illustrating the two droplets is the symmetry plane; the bottom surface is the condensing surface; and the remaining faces are open atmosphere. As illustrated in Fig 3, atmospheric values for volume fraction, t, U and P are kept at zero gradient.

Boundary conditions

Table 1: Shows boundary conditions for the simul-ation domain for the present investigation (Initial Boundary conditions).

2.3 Numerical simulation details

In order to investigate the effect of surface wett-ability, number of simulations (series A) with diffe-rent parameters such as receding angle, advancing angle and various radii of droplets have been used as shown in Table 2. 

In the present work simulations, a radius of 12.5μm for the first droplet kept constant throughout the investigation. Three sets of simulations were per-formed 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 droplets 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

3.1 Effect of different surface wettability and dif-ferent 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 differ-ent 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 suffi-cient 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 con-densing surface. A receding angle of 107° and an advancing angle of 117° creating a hydrophobic con-dition 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 com-pleted and applicable data was obtained. Results obtained in the present work showed varying wett-ability 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).

3.2 Effect of different surface wettability with dif-ferent 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 wettabilitys 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 utilis-ing 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 mer-ges together and forms singular droplet with larger mass and condense gradually. However, small differ-ences 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 conden-sing 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 con-sidered 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.

3.3 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.

3.3.1 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 conden-sation 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. 

3.3.2 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, super-hydrophobic 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 

Fig 27 and 28 shows the results for cases with drop-lets radii of 12.5um & 50um, 12.5um and 100um for the same wettability parameters discussed in Fig 26.

3.4 Effect of different wettability characteristics on condensation rates of the phenomenon with a dif-ferent 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.

3.4.1 Effect on the volume integral of condensation rate of droplets with different Radii

Fig (29-34) shows the effect of gravity vector on the droplets coalescing on surfaces with different wett-ability characteristics. For this, cases 1-9 from series (A) with altered wettabilitys 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 phe-nomenon are much more important for the considered spatial and temporal scale (Uddin et al., 2021).

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. (Geor-goulas et al., 2017). This modified solver was then applied to carry out series of simulations of the process of droplet coalesence in dropwise conden-sation. 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 quanti-tative 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 con-tinued to condense. In other cases, due to momentum created by merging, droplets oscillated and lifted off from the condensing surface. These two events occur-red 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 wett-ability 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 conden-sation.

It gives me immense pleasure to have completed this dissertation and acknowledge the support and guid-ance 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.

The authors declared that there is no conflict of interest in this article.