In recent years, established heat transfer models for dropwise condensation (DWC) have been refined to consider the influence of wetting behavior, surface structure and nucleation dynamics on the heat transfer rate in more detail. Despite these efforts to develop more sophisticated models, uncertainties of the model parameters still lead to a h igh variation of the calculated heat transfer rate. In this study, we apply quantitative sensitivity analysis to a pure steam DWC heat transfer model in order to attri bute the variation of the model result to its input parameters. Four scenarios with different variations of the model parameters are discussed and sensitivity coefficients for each parameter are calculated. Our results sho w that the contact angle and the nucleation site density have the greatest influence on the mode l result if no additional coating layer is considered. The influen ce of the nucleation site densi ty is mainly due to the large uncertainties associated with this parameter. In scenarios with an additional coating layer, the heat flux is mainly governed by the thickness of the coating la yer, underlining th e need for thin conformal coatings to effectivel y promote DWC. Furthermore, tre nds within the heat transfer model are discussed and beneficial conditions for a high heat f lux are identified for each scenario. The results underline t hat the wetting properties of functional surfaces should be tailored towards medium contact angles in the range of 70° to 1 30° and low contact angle hystereses below 10° to ach ieve high heat flux DWC. Keywords: dropwise condensation, heat transfer enhancement, sen sitivity analysis, nucleation 2 Nomenclature ݋ܤ Bond number ॱሺ⋅ሻ expected value of argument ሺ⋅ሻ ݃ gravitational acceleration, m s-2 ݆ base sample size ܬ sample size ݇ number of model input parameters ݈ୡ capillary length, m ݊ small drops size distribution, m-3 ܰ large drops size distribution, m-3 ܰୱ nucleation site density, m-2 ݌ pressure, kPa ݍሶ heat flux, W m-1 ܳሶୢ heat flow rate through a single drop, W ݎ drop radius, m ݎୣ effective drop radius, m ݎ୫ୟ୶ maximum drop radius, m ݎ୫୧୬ minimum drop radius, m ܴ୥ specific gas constant, J kg-1 K-1 ܵ୘ total effects sensitivity index ܶୱୟ୲ saturation temperature, K Varሺ⋅ሻ variance of argument ሺ⋅ሻ ܺ௜ model input parameter ܆∼௜ ܬൈሺ݇െ1ሻ matrix of all parameters but ܺ௜ ܻ model result Greek symbols ߙ∗ equivalent heat transfer coefficient, W m-2 K-1 ߙ୪୥ interfacial heat transfer coefficient, W m-2 K-1 ߜୡ coating thickness, μm Δ݄୴ latent heat of vaporization, kJ kg-1 Δܶୡ temperature difference due to the conduction through the coating layer, K Δܶୡ୳୰୴ temperature difference due to drop curvature, K Δܶୢ temperature difference due to the conduction through the drop, K Δܶ୧ temperature difference due to interfacial resistance, K Δܶୱ୳ୠ subcooling temperat ure difference, K 3 Δߠ contact angle hysteresis, deg ߠ contact angle, deg ߠୟ advancing contact angle, deg ߠ୰ receding contact angle, deg ߣୡ thermal conductivity o f the coating, W m-1 K-1 ߣ୪ thermal conductivity of the condensate, W m-1 K-1 ߩ୥ vapor density, kg m-1 ߩ୪ liquid density, kg m-1 ߪ୪୥ surface tension, N m-1 ߬ sweeping period, s Subscripts c coating d drop g vapor phase l liquid phase lg liquid vapor interphase Abbreviations DWC dropwise condensation PTFE Polytetrafluoroethylene 4 1. Introduction The potential for an enhanced heat transfer compared to filmwis e condensation has motivated extensive research on dropwise condensation (DWC) for over 90 y ears [1,2]. Beside the question of how to promote this s uperior condensation mode in h eat exchangers, the modeling and calculation of the heat flux during DWC is a central topic of this research. In 1966, Le Fevre and Rose published a fundamental model for the heat trans fer during DWC [3]. In this model, a characteristic density distribution of the drop sizes is calculated and then combined with the size-dependent heat flow rate through a single drop. T his conceptual modeling approach continues to be the basis for a majority of the models for DWC [4]. Both aspects of the model – the heat flow rate through a single drop and the dr op size distribution – have been intensively discussed and refined in numerous publications. Imp ortant developments in DWC modeling include the application of population balance methods t o c a l c u l a t e t h e d r o p s i z e distribution for small drops [5,6] , the consideration of non-he mispherical drops with contact angels over 90° [7] and the modeling of the heat transfer on ad vanced functional surfaces like superhydrophobic nanostructured surfaces [8–11] and lubricant-i nfused surfaces [10,12,13]. Extensive reviews are availabl e on the topic of DWC modeling [4 ,14–16]. Despite these considerable effort s, model results often deviate from experiments. Recently, Wang et al. have compared the model results of typical DWC heat transfer models with experimental data from the liter ature and found that existing m odels are not generally applicable to various conditions during DWC [17]. Similar comparisons of m odel results and experimental data have been used by other researchers to validate heat trans fer models for DWC [6,7,18–20]. If significant deviations between model and experimental result s occur, they are often explained with a lack of detail in the investigated heat transfer model o r with experimental errors like inaccurate heat transfer measurements and the presence of non-c ondensable gases [2,14]. While 5 both of these explanations can b e valid, deviations can also re sult from inaccurate values for the model input parameters. Naturally, all DWC heat transfer models rely on measurements, e stimations and assumptions regarding their input parameters . All values of the model input parameters are therefore associated with an uncertainty. Some input parameters of the DW C models are notoriously difficult to measure or calculate. This is especially true for the parameters associated with the nucleation process of new drops. Most of the DWC models calcula te radius of newly formed drops using classical nucleation theory. It is then assumed tha t new drops form at specific nucleation sites on the surface as soon as they are not occupie d by another drop. These new drops are very small and grow rapidly during condensation, whic h makes any direct observation of the process difficult. Thus, ex perimental results to support the assumptions regarding nucleation during DWC are scarce. At the same time, it is often not possible to reliably calculate all the relevant model parameters from theory. Classical nuclea tion theory is known to be unreliable in predicting experime ntal results in other cases su ch as homogeneous nucleation of argon [21–23]. Despite these known issues, the uncertainties as sociated with model input parameters like the nucleation site density are rarely consider ed when modeling DWC, even though they can be significant a nd may lead to a high variation in the calcula ted heat flux. In order to systematically refine and validate existing models, it is essential to know how the variation of the model result can be attributed to each of the model input parameters. The parameters that have the greatest influence on the model result should be identified so that researchers can focus their modeling and experimental efforts o n improving the data basis for DWC modeling where necessary. In this theoretical study, we app ly variance based quantitative sensitivity analysis to a DWC heat transfer model in order to e stablish a ranking of the model input parameters based on their total sensitivity indices in fo ur different scenarios. The scenarios are defined with regar d to physical plausibility (sce narios 1 and 2) and information 6 from measurements (scenarios 3 and 4). In addition to the quant itative sensitivity analysis, we identify and discuss regions in the input parameter space for e ach scenario that lead to high heat fluxes according to the DWC heat transfer model. 2. Dropwise condensation model 2.1 Heat transfer model Starting point for the modeling of the heat flow in this study is the model by Kim and Kim published in 2011 [7]. This model follows the modeling approach by Le Fevre and Rose [3] and is suitable as a point of reference since it represents a s ignificant extension of previous models by including non-hemispherical drops. At the same time, the model is formulated in general terms, so that its application is not limited to specif ic surfaces. The heat flux ݍሶ during DWC is calculated from equation (1) ݍሶൌܳ׬ሶ ୢሺݎሻ݊ሺݎሻdݎ௥౛ ௥ౣ౟౤൅ܳ׬ሶୢሺݎሻܰሺݎሻdݎ௥ౣ౗౮ ௥౛ ( 1 ) where ܳሶୢሺݎሻ is the heat flow rate through a single drop of the radius ݎ ,݊ሺݎሻ is the drop size distribution for small drops with a radius between the minimum drop radius ݎ୫୧୬ and the effective drop radius ݎୣ and ܰሺݎሻ is the drop size distribution for large drops with radii up to the maximum drop radius ݎ୫ୟ୶. As derived by Kim and Kim [7], the heat flow rate through a single drop is given by equation (2) ܳሶୢൌ୼்౩౫ౘగ௥మቀଵିೝౣ౟౤ ೝቁ భ మഀౢౝሺభషౙ౥౩ഇሻାೝഇ రഊౢ౩౟౤ഇାഃౙ ഊౙ౩౟౤మഇ ( 2 ) where Δܶୱ୳ୠ is the subcooling temperature, ߠ is the contact angle, ߣ୪ is the thermal conductivity of the condensate, ߜୡ is the coating layer thickness and ߣୡ the thermal conductivity of the coating. The interfacial heat transfer coefficient ߙ୪୥ is calculated from fluid properties (specific 7 gas constant ܴ୥, saturation temperature ܶୱୟ୲, latent heat of vaporization Δ݄୴ and vapor phase density ߩ୥) using equation (3) [24]. ߙ୪୥ൌଶ ඥଶగோౝ்౩౗౪ ୼௛౬మఘౝ ்౩౗౪ ( 3 ) Since the primary mechanism of d rop growth change s depending on the drop size , two regions are distinguished in the drop si ze distribution: very small dro ps that grow primarily by direct condensation on the drop surface and larger drops that grow pri marily by coalescence with other drops [25,26]. For large drops, the drop size distribution is c alculated according to the equation given by Le Fevre and Rose [3] as a function of the maximum dro p radius that is occurring on the surface before the drop is removed by gravity or external f orces. ܰሺݎሻൌଵ ଷగ௥మ௥ౣ౗౮ቀ௥ ௥ౣ౗౮ቁିଶ/ଷ ( 4 ) The drop size distribution for small drops is depending on nucl eation processes during condensation. It is calculated based on the population balance method [6] using the following equations [7]. ݊ሺݎሻൌଵ ଷగ௥౛య௥ౣ౗౮ቀ௥ ௥ౣ౗౮ቁିଶ/ଷ ௥ሺ௥౛ି௥ౣ౟౤ሻ ௥ି௥ౣ౟౤஺మ௥ା஺య ஺మ௥౛ା஺యexpሺܤଵ൅ܤଶሻ ( 5 ) ܣଵൌ୼்౩౫ౘ ଶఘౢ୼௛౬ ( 6 ) ܣଶൌఏሺଵିୡ୭ୱఏሻ ସఒౢୱ୧୬ఏ ( 7 ) ܣଷൌଵ ଶఈౢౝ൅ఋౙሺଵିୡ୭ୱఏሻ ఒౙୱ୧୬మఏ ( 8 ) 8 ܤଵൌ஺మ ఛ஺భቂ௥౛మି௥మ ଶ൅ݎ୫୧୬ሺݎୣെݎሻെݎ୫୧୬ଶlnቀ௥ି௥ౣ౟౤ ௥౛ି௥ౣ౟౤ቁቃ ( 9 ) ܤଶൌ஺య ఛ஺భቂݎୣെݎെݎ ୫୧୬lnቀ௥ି௥ౣ౟౤ ௥౛ି௥ౣ౟౤ቁቃ ( 1 0 ) ߬ൌଷ௥౛మሺ஺మ௥౛ା஺యሻమ ஺భሺଵଵ஺మ௥౛మିଵସ஺మ௥౛௥ౣ౟౤ା଼஺య௥౛ିଵଵ஺య௥ౣ౟౤ሻ ( 1 1 ) The drop size distribution for small drops is therefore dependi ng on the fluid properties, the minimum drop radius ݎ୫୧୬, the effective drop radius ݎୣ, the contact angle ߠ , as well as the coating thickness ߜୡ and heat conductivity of the coating layer ߣୡ. The minimum drop radius is in turn calculated from fluid properties and surface subcooling based on classical nucleation theory using equation (12) ݎ୫୧୬ൌଶ்౩౗౪ఙౢౝ ୼௛౬ఘౢ୼்౩౫ౘ ( 1 2 ) where ߪ୪୥ is the surface tension and ߩ୪ is the density of the condensate. In the model by Kim and Kim [7], the subcooling temperature Δܶୱ୳ୠ includes the temperature drop due to interfacial resistance Δܶ୧, the temperature drop due to drop curvature Δܶୡ୳୰୴, the temperature drop due to conduction through the drop Δܶୢ as well as the temperature drop due to the conduction through the coating layer Δܶୡ. Δܶୱ୳ୠൌΔ ܶ୧൅Δܶୡ୳୰୴൅Δܶୢ൅Δܶୡ ( 1 3 ) We adopt this definition since it simplifies the calculation of the heat flux with equation (1). With regard to equation (12) it should be noted that, by includ ing the temperature drop due to the conduction through the coating layer, this definition of th e subcooling temper ature leads to 9 an underestimation of the minimum drop radius. The interrelatio n between the minimum drop radius and the thermal resistan ce of the coating is discussed i n detail elsewhere [27]. For the effective drop radius, the following relationship with the nucleation site density ܰୱ follows from a simplified geometric consideration that assumes an arrangement of the nucleation sites in form of a square array [7]. ݎୣൌሺ4ܰୱሻିଵ/ଶ ( 1 4 ) The maximum drop radius on a vertical wall is calculated based on the force balance between surface tension and gravity using the following equation [7]. ݎ୫ୟ୶ൌቀ଺஢ౢౝሺୡ୭ୱఏ౨ିୡ୭ୱఏ౗ሻୱ୧୬ఏ గఘౢ௚ሺଶିଷୡ୭ୱఏାୡ୭ୱయఏሻቁଵ/ଶ ( 1 5 ) To reduce the number of model input parameters for the sensitiv ity analysis, the advancing contact angle ߠୟ and the receding contact angle ߠ୰ are calculated from the contact angle hysteresis Δߠ based on the assumption that the contact angle is the arithmet ic mean of the two values: ߠୟൌߠ൅୼ఏ ଶ ( 1 6 ) ߠ୰ൌߠെ୼ఏ ଶ ( 1 7 ) Using equations (2) – (17), the he at flux is calculated from eq uation (1) for a gi ven subcooling temperature. The aim of this study is to evaluate the influence of the model parameters on the heat transfer efficiency calculated by the model. Therefore, th e subcooling temperature is not considered as a model parameter. Instead, an equivalent heat tr ansfer coefficient ߙ∗ is defined in equation (18) and calculated f rom two subcooling temperature s and their corresponding heat fluxes. 10 ߙ∗ൌଵ ଶ൬௤ሶభ ୼்౩౫ౘ,భ൅௤ሶమ ୼்౩౫ౘ,మ൰ ( 1 8 ) Subcooling temperatures of Δܶୱ୳ୠ,ଵൌ1 K and Δܶୱ୳ୠ,ଶൌ 10 K are used for the sensitivity analysis. According to the defin ition of the subcooling tempera ture in equation (13), the equivalent heat transfer coefficient includes the heat transfer from the fluid to the surface of the coating layer as well as the thermal conduction through the coa ting layer. 2.2 Dropwise condensation criterion The heat transfer model allows for a calculation of the mean he at transfer coefficient for arbitrary combinations of input p arameters. However, the model provides no info rmation as to whether DWC actually occurs with the given input parameters. Re cently, Cha et al. have proposed a criterion for dropwise to filmwise transition based on the Bond number Bo of the largest drop immediately before departure [28]. We adopt this c riterion and assume that DWC occurs only if Bo ≤ 1.4. The Bond number for a drop in equilibrium immediately b efore departure is defined as follows. ݋ܤ ൌ ௥ౣ౗౮మ ௟ౙమ ( 1 9 ) The capillary length ݈ୡ is defined as ݈ୡൌටఙౢౝ ఘౢ௚ ( 2 0 ) where ݃ is the gravitational acceleration. Using equations (15), (19) and (20) the Bond number can be calculated based on t he contact angles alone. ݋ܤ ൌ଺ሺୡ୭ୱఏ౨ିୡ୭ୱఏ౗ሻୱ୧୬ఏ గሺଶିଷୡ୭ୱఏାୡ୭ୱయఏሻ ( 2 1 ) 11 Dropwise to filmwise transition due to nucleation site saturati on [29] is not considered in this work since it depends heavily on the surface subcooling tempera ture, which is not investigated in the sensitivity analysis. 2.3 Input parameters With regard to the equations above, the following six input par ameters are required to calculate the heat flux for a given subcoo ling temperature with equation (1):  contact angle ߠ  contact angle hysteresis Δߠ  the fluid properties of the sa turated steam and condensate, whi ch can be calculated from the pressure ݌ for a specific fluid  the nucleation site density ܰ ୱ  the thickness of the coating ߜୡ  the thermal conductivity of the coating ߣୡ In order to calculate the large number of model results require d for the sensitivity analysis, the model is implemented in the python programming language. For th e sensitivity analysis, pure water is considered as the conde nsing fluid. All fluid properti es are calculated using the CoolProp library [30]. 3. Variance based global sensitivity analysis 3.1 Sampling method and sensitivity index The aim of the sensitivity analy sis is to attribute the variati on of the equivalent heat transfer coefficient calculated with the DWC heat transfer model to the input parameters of the model [31]. Variance based sensitivity analysis allows this attributi on to be quantified in form of a 12 sensitivity index with a numeri c value between 0 and 1 for each input parameter. In this study, the total effect sensitivity indices ܵ୘, as introduced by Homma and Saltelli [32], are calculated for each input parameter according to the method of Sobol’ [33] . The total effect sensitivity index provides a measure for the overall effect of each input p arameter on the model results, including direct effects as well as interactions with all other input parameters [32]. In this study, the aim of the sensitivity analysis is to establish a ranking (or factor priorization ) [31,33] of the input parameters of the DWC heat transfer model with regard to their relative importance in determining the model result. The uncertainties of input parame ters with high total effect sensitivity indices affect the model result the most, while inp ut parameters with a low total effect sensitivity index have only little influence on the vari ation of the model result. In order to calculate the total effect sensitivity indices, an input par ameter space is defined by choosing a value range for each input parameter. The value ranges are ch osen based on their physical meaning, information from the literature or previous measuremen ts. A number of samples are then generated from the input parameter space by using the samp ling strategy established by Saltelli et al. [34,35]. A base sample size of j = 1000 is chosen, resulting in J samples for each scenario, depending on the number of input parameters ݇ [34]: ܬൌ݆ ሺ 2 ݇൅2 ሻ ( 2 2 ) For these samples, the equivalent heat transfer coefficients ar e calculated with the DWC heat transfer model. The total effect sensitivity index ܵ୘,௜ for a given model input parameter ܺ௜ with ݅ൌ1 ,…,݇ is defined in equation (22) [35] ܵ୘,௜ൌ1െ୚ୟ୰ ሾॱሺ௒ ∣ ܆∼೔ሻሿ ୚ୟ୰ ሾ௒ሿ ( 2 3 ) where ܻ is the model result (in this case the equivalent heat transfer coefficient) and ܆∼௜ is a ܬൈሺ݇ െ1ሻ matrix of all model input parameters except for ܺ௜. The total effect sensitivity 13 indices are then calculated from the input parameter samples an d their corresponding model results according to the method of Sobol’ [33]. For the samplin g of the input parameter space as well as for the calculation of the sensitivity indices, the python library SALib [36] is used. The python code used to produce th e results and figures in this paper is available online [37]. 3.2 Definition of input parameter space The sensitivity of the model with respect to the various input parameters is largely determined by the value ranges of the input parameters. Depending on the d efinition of the value ranges, different statements can be made about the model based on the s ensitivity analysis. Two main aspects are considered seperately in the definition of the inpu t parameter space: 1) the influence of the coating and 2) the influence of the range of the input p arameters. This results in four scenarios for the sensitivity analysis. In scenario 1 and 2, a wide range of input parameters is considered without any coating a nd with an additional coating l ayer, respectively. Scenario 3 and 4 are defined with a more narrow range of the input paramet ers, again without (scenario 3) and with (scenario 4) an additional coating layer. In the follo wing, the four scenarios are defined and their motivation is explained in detail. Table 1 provides a n overview over the input parameter space for all scenarios. Table 1 Input parameter space for all four scenarios. ࣂ / deg ࢖ / kPa ܛࡺ / m-2 ઢࣂ / deg ܋ࢾ / μm ܋ࣅ / (W m-1 K-1) Scenario 1 5 … 175 5 … 500 109 … 1015 1 … 90 no coating no coating Scenario 2 5 … 175 5 … 500 109 … 1015 1 … 90 0.1 … 50 0.2 … 10 Scenario 3 83 …. 93 11 … 13 109 … 1015 24 … 44 no coating no coating Scenario 4 83 …. 93 11 … 13 109 … 1015 24 … 44 0.1 … 1 5 … 10 14 In scenario 1, DWC on an uncoated surface is assumed. The input parameters (contact angle, contact angle hysteresis, nucleat ion site density and pressure) are varied over a wide range. Therefore, a wide spread of the model results can be expected. The purpose of this scenario is to investigate the maximum range f or the calculated equivalent heat transfer coefficients and to investigate basic correleations between the input parameters an d the model results. The value ranges are chosen based on the criterion of physical and techni cal plausibility. For the contact angle the whole range from very hydrophilic ( ߠൌ5 ° ሻ to very hydrophobic ሺߠ ൌ 175°ሻ is considered. The contact angle hys teresis is varied from 1° to 9 0°. Due to the wide range of values for both the contact angl e and the contact angle hystere sis, equations (16) and (17) give advancing contact angles of ߠୟ൐ 180° or receding contact angles of ߠ୰൏0 ° in some cases. In those cases, ߠୟൌ 180° or ߠ୰ൌ0 ° is assumed, respectively. These combinations of contact angle and contact angle hysteresis are still used for the calcu lation of the sensitivity indices but not shown in the scatterplots in section 4.2. For the vapor pre ssure, a value range of 5 kPa to 500 kPa is selected against the background of possible technica l applications. The value range of the nucleation site density is defined as 10ଽ mିଶ൏ܰୱ൏1 0ଵହ mିଶ based on values from the literature gi ven in Table 2. 15 Table 2 Orders of magnitude for t he nucleation site density ܰୱ from the literature. ܛࡺ / m-2 Type of surface or promoter and reference 10ଽ Fluorinated thiol coating [9] 10ଵ଴ Fluorinated silane coating [9], hydrophobic polymer film and a silanized glass [38] 10ଵ଴…10ଵଵ Various metal surfaces [39] 10ଵଵ Superhydrophobic surface [7], lubricant infused surface [12], PTFE and dio ctadecyl disulphide [6] 10ଵଵ…10ଵଶ Sol-gel coating on aluminum substrate [40] 10ଵଶ Copper with dioctadecyl disulphide promoter [41] 10ଵସ Various functionalized titanium surfaces [18] 10ଵଷ…10ଵହ Theoretical study [42] Scenario 2 corresponds to scenar io 1 and additionally, the infl uence of a coating layer is taken into account. The parameters of the coating layer are varied in a wide range of values of 0.1 μm to 50 μm for the coating thickness and 0.2 W m -1 K-1 to 10 W m-1 K-1 for the thermal conductivity of the coating. The ranges are defined based on te chnical considerations: coatings with a thickness below 0.1 μm tend to be not robust enough for technical applications. With regard to the thermal resistan ce, the coating thickness was lim ited to 50 μm. The value range for the thermal conductivity coefficients was determined based on potential coating materials (e.g. approximately 0.2 W m-1 K-1 for PTFE and 10 W m-1 K-1 for composite coatings [43]). In Scenario 3, DWC on an uncoated surface is considered. In con trast to scenario 1, the input parameter space is more narrowly defined. Previous condensation experiments on a mirror- finished stainless steel sample serve as orientation. The value ranges were chosen to reflect the uncertainty associated with the measurement of each input param eter. To our knowledge, there is currently no experimental procedure available to measure the nucleation site density directly. Instead of a direct measurement, the values for this model para meter have to be estimated based on data from the literature. Therefore, the same range of value s as in scenario 1 and 2 is used. 16 The aim of the sensitivity analysis with this scenario is to co nsider the effects of the measurement uncertainties of all input parameters on the model result. This is intended to make a statement as to which input parameters should be determined m ore precisely in experiments in order to better va lidate the model. Scenario 4 corresponds to scenario 3, but the additional influe nce of a coating layer is considered. The range of values is selected in such a way that the coatings have beneficial properties for the heat transfer (small coating thickness in th e range of range of 0.1 μm to 1 μm and high thermal conductivit y in the range of 5 W m-1 K-1 to 10 W m-1 K-1). 4. Results and discussion 4.1 Quantitative sensitivity analysis Deviations between the heat transfer model and DWC experiments can result from the model structure as well as from uncertainties associated with the mod el input parameters (which might be measured or calculated). In m any studies, the uncertainty of the model input parameters is not considered as part of the model validation. To assess, whic h model input parameters have the greatest influence on the va riation of the model result, th e total effect sensitivity indices are calculated for the four DWC scenarios. The total effect sensitivity indices are shown in Fig. 1. Overa ll, it is clear that in the scenarios with a coating layer (scenario 2 and scenario 4), the thermal r esistance of the coating (thermal conductivity and coating thickness) has a cru cial influence on the calculated heat transfer coefficient. This also applies to the case of a coating with a low coating thickness and comparatively high thermal conduc tivity (scenario 4). In additi on to the thermal conductivity of the coating, the model result depends above all on the conta ct angle (scenarios 1 and 2) and the nucleation site densi ty (scenarios 3 and 4). 17 Fig. 1. Total effect sensitivity indices ܵ୘ for the four scenarios. Scenarios 1 and 2 represent a wide input parameter space without and with a coating layer, re spectively. Scenarios 3 and 4 represent a more narrow input parameter space without and with a coating layer, respectively. Model input parameters are: contact angle ߠ ,pressure p, nucleation site density ܰୗ, contact angle hysteresis Δߠ ,coating thickness ߜୡ and thermal conductivity of the coating ߣୡ. Error bars indicate the 0.95 confidence level. The influence of the nucleation site density on the model resul t is particularly strong in scenario 3. In this scenario, the input parameter space is narrowed down to reflect the information that can be gained by DWC experiments and contact angle measurements . In the case of the nucleation site density, there are no additional information ob tainable from direct measurements, which leads to the high sensitivity index of the nucleation sit e density observed in scenario 3. This means that, due to the high uncertainty associated with th e nucleation site density, the heat flux calculated by the model can deviate from experimental resu lts even if the heat transfer model itself is perfectly accurate and the experiments are cond ucted with great care. 18 The high sensitivity index of the nucleation site density in sc enario 3 also offers a starting point for the experimental determinati on of the nucleat ion site densi ty. Using model calibration [31] the nucleation site density could be estimated from experimenta l data if the other model parameters, the heat flux and the subcooling temperature are me asured accurately. Ideally, these experiments are performed on unc oated or very precisely coated samples in order to minimize the influence of the coating laye r as investigated in scenario 4. A prerequisite for this experimental determination of the nucleation site density is th at the heat transfer model represents the actual processes during DWC with sufficient accu racy. In particular, additional effects on the heat tra nsfer, that are not represented in the m odel (e.g. non-condensable gases), must not influence the experiments. Under these conditions, mod el calibration can be a useful approach for the investigation of nucleation phenomena in DWC a llowing validation or adaptation of the previous models. The total effect sensitivity indices provide a quantitative mea sure for the influence of each input parameter on the model result. However, they do not indicate an y trends between the values of the input parameters and the model result. In order to identify regions in the input parameter space that lead to a high equivalent heat transfer coefficient, a qualitative sensitivity analysis of the DWC model is conducte d in the following section. 4.2 Qualitative sensitivity analysis 4.2.1 Scenario 1 In this scenario, the input values for the model are varied ove r a very wide range, not taking into account the insulating effect of a coating. Under these co nditions, the calculated heat transfer coefficient is most dependent on the contact angle of the condensate drops, while the 19 pressure, nucleation site dens ity and contact angle hysteresis show low total effect sensitivity indices of less than 0.2. Fig. 2 shows the influence of contact angle and pressure on the calculated heat transfer coefficient. Both influencing pa rameters show a clear tendency: The contact angle significantly reduces the heat transfer when it takes on very large or very s mall values. Very high heat transfer coefficients of over 1300 kW m -2 K-1 occur at medium contact angles between 70° and 130°. Over the entire range of contact angles, a higher pressur e has a positive effect on the calculated heat transfer coefficients. Fig. 2. Scenario 1 - equivalent heat transfer coefficient ߙ ∗ vs. contact angle ߠ and pressure p. Not all of the data points shown in Fig. 2 fulfill the DWC crit erion of Bo ≤ 1.4. Many data points in the hydrophilic region ha ve high Bond numbers and fil mwise condensation is expected for these combinations of input p arameters. Fig. 3 shows only t he data points for which DWC 20 is expected. It can be seen that DWC on hydrophilic surfaces wi th contact angles below 90° is only possible if the contact angle hysteresis is low enough. A very low contact angle hysteresis below 10° is necessary to achieve very high heat transfer coeff icients of over 1300 kW m-2 K- 1. Fig. 3. Scenario 1 - equivalent heat transfer coefficient ߙ∗ vs. contact angle ߠ and contact angle hysteresis Δߠ ,only data points with Bo ≤ 1.4 are shown. 4.2.2 Scenario 2 This scenario corresponds to s cenario 1, but additionally the i nsulating effect of a coating layer is considered. The thermal condu ctivity and the coating thickne ss are varied over a wide range. In this scenario, the layer th ickness has the strongest influen ce on the calculated heat transfer coefficients. This is followed by the contact angle and the the rmal conductivity of the coating. 21 Contact angle hysteresis, nucleation site density and pressure have a very small influence compared to these parameters. Fig. 4 shows the influence of c ontact angle and coating thickne ss on the equivalent heat transfer coefficient. As in the previous scenario 1, it can be seen that the highest equivalent heat transfer coefficients are achieved at medium contact angles (indicated b y dark points in the diagram). High equivalent heat transfer co efficients of more than 500 kW m -2 K-1 are only achieved in this scenario with very low coa ting thicknesses of less than 0. 3 μm. Fig. 4. Scenario 2 - equivalent heat transfer coefficient ߙ∗ vs. coating thickness ߜୡ and contact angle ߠ. 4.2.3 Scenario 3 In this scenario, no coating is considered. The other input par ameters are varied in a narrow range to reflect the information obtainable by measurements. Th e nucleation site density cannot 22 be measured directly and is therefore associated with large unc ertainties. This is also evident in the sensitivity coefficients of the model parameters: the nucle ation site density has the greatest influence on the calculated heat transfer coefficients, by a cl ear margin to all other parameters. Fig. 5 shows the infl uence of the nucleation site density and t he contact angle hysteresis on the heat transfer. A clear trend can be seen for both variables: wi th increasing nucleation site density and decreasing contact angle hyst eresis, the heat transfer coef ficient increases. The Bond numbers in this scenario range b etween 0.3 and 0.9, therefore t he DWC criterion is satisfied for all datapoints. Fig. 5. Scenario 3 - equivalent heat transfer coefficient ߙ ∗ vs. nucleation site density ܰୱ and contact angle hysteresis Δߠ. 23 4.2.4 Scenario 4 This scenario corresponds to scenario 3, and additionally the i nfluence of a coating layer is considered. Based on the results from scenario 2, a thin coatin g with thicknesses of less then 1 μm and with high thermal conductivity of 5 W m -1 K-1 to 10 W m-1 K-1 is chosen, so that high heat transfer coefficients are possible in principle. However, the model result in this scenario is still largely determined by the coating thickness, followed by the nucleation site density. Fig. 6 shows the influence of layer thickness and nucleation si te density. The trends for these two parameters described in scen arios 2 and 3 are confirmed in this scenario: the highest equivalent heat transfer coefficients occur with higher nucleat ion site densities and smaller coating thicknesses. Overall, the calculated heat transfer coef ficients are in a similar range as in scenario 3. Due to the additional insulating effect of the c oating, the value s scatter more strongly in the direction of low er equivalent heat transfer coe fficients, which is particularly evident at high nucleation site d ensities. Similar to scenario 3, the Bond number criterion of Bo ≤ 1.4 is satisfied for all datapoints. 24 Fig. 6. Scenario 4 - equivalent heat transfer coefficient ߙ∗ vs. nucleation site density ܰୱ and coating thickness ߜୡ. 5. Conclusions In this study we presented quant itative and qualitative sensiti v i t y a n a l y s i s o f a D W C h e a t transfer model. Our findings allo w to attribute the variation o f the model result to each of the model input parameters in four different scenarios. These resul ts can serve as a guideline for future model development and validation as well as for the desi gn of DWC experiments. Using variance-based quantitativ e sensitivity an alysis we found that the uncertain ties in regard of the nucleation site density ca n have a significant influence on the calculated model result. When the model parameters are varied over a very wide range (sc enarios 1 and 2), the contact angle has a great influence on the calculated heat transfer coe fficient. Qualitative sensitivity analysis showed that medium contact angles between 70° and 130° are optimal to achieve high 25 heat transfer coefficients according to the model. The contact angle hysteresis has less influence on the heat transfer coefficient but low values of less than 10 ° are preferable. In any case, the contact angle hysteresis must b e low enough to facilitate DWC a ccording to the Bond number criterion proposed by Cha et al . [28], especially on hydrophili c surfaces. In scenario 3, which is considering a narrow input parameter sp ace without any coating, it becomes clear that the variation of the model result can be att ributed mainly to the nucleation site density. The reason for this is the high uncertainty assoc iated with this parameter, which reflects the difficulties in reliably measuring or calculating the nucleation site density. The quantitative sensitivity analysis illustrates the high relevanc e that this aspect has for future DWC research. Considering the high sensitivity index of the nuc leation site density, model calibration can be a viable approach to determine the value of this parameter experimentally and to allow for a closer inve stigation of nucleation phenomena in DWC. Our results further underline the importance of minimizing the thermal resistance of any coating layer that is used to promote DWC. In the scenarios with a coat ing layer (scenarios 2 and 4), the coating thickness is the inpu t parameter with the highest s ensitivity index. This is the case, even if thin coatings with a coating thickness of less than 1 μ m and a high thermal conductivity in the range of 5 W m-1 K-1 to 10 W m-1 K-1 are considered, as shown in scenario 4. In conclusion, the sensitivity analysis showed that the thermal resistance of the coating, the contact angle and the nucleation site density are the main para meters influencing the heat transfer during DWC. In order to maximize the heat flux during DWC, coatings should be as thin as possible, preferably less than 0.3 μm, and their wettin g properties should be tailored towards average contact angles b etween 70° and 130° with a low contact angle hysteresis of less than 10°. Further research is needed to provide experiment al data on the nucleation site density during DWC. 26 Declaration of Competing Interest The authors declare that they ha ve no known competing financial interests or personal relationships that could have a ppeared to influence the work re ported in this paper. Acknowledgements The authors would like to ac knowledge the financial support by the European Union and the Free State of Saxony (SA B grant number 100317611).