A nitrogen Leidenfrost droplet on a water pool: Experiments, theory and simulations of droplet shrinkage and ice formation

The cooling capabilities of liquid nitrogen are exploited in various ﬁelds, where the liquid is often in apparent contact with a soft or even liquid partner. Even though the heat transfer between the partners is mostly of actual concern for the given application, it is not yet fully described. In the present work, the shrinkage of a nitrogen Leidenfrost droplet on a water pool and the resulting formation of ice inside the pool are examined experimentally, theoretically and numerically. Experiments are performed using nitrogen droplets of varying size, deposited onto a water pool which is initially always at its melting temperature. Droplet shrinkage and ice formation are captured using a high-speed video camera providing a synchronized top and side-view on the scene. An existing analytical model for the interface shape of the droplet and the pool in the given situation is extended to enable theoretical prediction of the temporal evolution of the droplet size and the volume of ice formed inside the pool. While only heat transfer at the droplet bottom is considered in the original model, in the present work also the major contributions to the heat transfer at the droplet top and at the pool meniscus are accounted for. Additionally, numerical simulations of droplet shrinkage are performed using a commercial ﬁnite-element simulation software ( COMSOL Multiphysics ). For both the theoretical model and the numerical simulations, the droplet and the pool are assumed isothermal and heat transfer is only considered in the gaseous ambient. Finally, both the theoretical predictions and numerical results generally well resemble the experimental ﬁndings for droplet shrinkage, where the numerical simulations show a slightly better agreement. Also the theoretical predictions for the ice volume forming inside the pool are in good agreement with the experimental results, conﬁrming the good predictive capabilities of the theoretical model for the present situation.


Introduction
Cryogenic liquids are used in various fields ranging from medicine [1][2][3][4] and food industry [5][6][7] over micro-electronics cooling [8,9] to machine processing and classical engineering [10][11][12][13].Due to their low saturation temperature (e.g.≈ 77 K for liquid nitrogen at atmospheric pressure), large temperature gradients and heat fluxes are generally involved when dealing with cryogenic liquids at atmospheric conditions.Also their evaporation is associated with significant heat transfer, making the use of cryogenic liquids favorable for many cooling or preservation applications in the mentioned fields.Due to the high evaporation rates, a cushioning layer of the liquid's vapor may form between a cryogenic liquid and other objects being at significantly higher temperatures.This vapor layer prevents actual contact between the cryogenic liquid and the warmer partner, which is commonly referred to as the Leidenfrost effect [14].As a consequence of the insulating effect of the tion.Moreover, in that case also solidification of the pool liquid may occur, which significantly changes the boundary conditions below the Leidenfrost object and thus, may finally add even more complexity to the entire scenario.
The dynamics of Leidenfrost droplets has been intensively studied over the last decades and even centuries.While the Leidenfrost effect for solid substrates has been the objective of intense research already since 1756 [14,[16][17][18], the situation for a liquid partner below the Leidenfrost object has been examined just in the last few decades [19][20][21][22][23].While mostly focusing on the droplet kinetics on the pool or on shape modeling of the droplet and the pool, the actual heat transfer between the involved partners has never been deeper examined in existing studies; although it is key for accurate modeling of the entire situation.As a consequence, the Leidenfrost effect for a liquid-liquid, i.e. droplet-pool system is not yet completely unraveled.In particular, the heat actually exchanged with the pool below, i.e. also the actual role of the ambient in the problem is not fully elucidated so far, especially for cryogenic temperatures.Moreover, solidification of the pool liquid below the droplet has not been examined, and particularly, it has not yet been used to provide insight into the heat transfer between the droplet and the pool.
In the present work, the dynamics of a liquid nitrogen droplet on a water pool is examined experimentally, theoretically and numerically with attention put on the processes taking place both on top and below of the pool meniscus.For the experiments, pure liquid nitrogen droplets are placed on the meniscus of a homogeneously temperature controlled water bath and the temporal evolution of the droplet size and the ice volume forming inside the pool below the droplet is captured using a high-speed video system in a top-view and side-view, respectively.Starting with a pool controlled to its melting temperature prior to an experiment, the amount of heat actually transferred between the droplet and the pool is reflected in the amount of ice present after thermal reequilibration of the pool following complete droplet evaporation.An existing theoretical model for the steady-state shape of a droplet and a pool in the Leidenfrost situation [21] is extended to enable prediction of the actual droplet shrinkage rate, which is finally used for transient modeling of the process.Moreover, also the heat transfer at the top of the pool is described in order to predict the total heat extracted from the pool during droplet evaporation.Finally, droplet shrinkage is also numerically simulated using a commercial finite-element simulation software (COMSOL Multiphysics).Apart from heat transfer and phase change in the pool, all relevant physical processes are accounted for in the simulations in order to numerically predict droplet shrinkage for the given situation.An overall comparison of the theoretical and numerical results with the experimental data reveals a very good agreement for both droplet shrinkage and growth of ice in the pool, indicating that all dominating physics are incorporated and accounted for in the theoretical model and the numerical simulations.

Methodology
Experiments and numerical simulations are performed in conjunction with the theoretical predictions for the situation of a nitrogen Leidenfrost droplet on a water pool.Both the experimental methodology and details about the numerical simulations are separately provided in the following.

Experimental
The experimental setup consists of 3 major sub-systems, namely a system for providing a well controlled water pool, a system for generating liquid nitrogen droplets and an observation system, from which the first and the latter are schematically shown in Figs. 1 and 2.An almost cubic water pool is provided in the center of a double-walled transparent acrylic glass box as shown in Fig. 1.An external liquid chiller (IKA, HRC 2 control) is used to supply a transparent cooling fluid (based on A droplet is placed on the pool meniscus provided in a circular cut-out of a thin kapton foil closing off the bottom of a top-insert extending down into the pool.The situation is observed from the top and the side as indicated through pictograms using a high-speed video system.(For interpretation of the colors in the figure, the reader is referred to the web version of this article.)silicon oil) between the double walls, resulting in a well controlled and homogeneous temperature of the pool being encapsulated by the cooling fluid.Using an insert to literally submerge the water meniscus into the liquid pool housing also results in a well controlled gaseous ambient around the meniscus being close to the pool temperature.The actual liquid meniscus is restricted through a thin kapton foil closing off the bottom of the insert but leaving a circular cut-out with a diameter of 5 mm.A syringe connected to the inner part of the pool is used to control the pool filling level and the resulting meniscus curvature through addition/draining of pool liquid.In order to reduce condensation on the outside of the pool, the entire system is encapsulated in styrofoam only leaving optical access for the observation system through cut-outs in the insulation at two opposed sides, and the top and bottom.An acrylic glass sheet is placed in each of the cut-outs for the side-view to maintain the side windows free from condensation.
A supply device for pure liquid nitrogen is used to deposit a nitrogen droplet on the submerged meniscus.Although an earlier development state of the device has been actually used for the present experiments, the basic working principle and functioning are described and characterized in detail in [24].Even though the resulting droplet size is not well controlled, the device allows deposition of liquid nitrogen droplets on the water pool in a drop-on-demand fashion without initial contamination of the droplet, e.g. with frost from the wet ambient.
Both the top-view on the Leidenfrost droplet and the side-view on the ice forming below the droplet are captured using a single highspeed video camera (Photron, FASTCAM NOVA S6).The camera with a long distance microscope (Navitar, 12X ZOOM) is combined with a customized lens attachment composed from opto-mechanical components (Thorlabs).It allows to capture two perspectives on the same scene, as indicated in Fig. 1 through pictograms.A schematic of the lens attachment and an unprocessed video frame captured using the optical system are shown in Fig. 2 a) & b), respectively.The small cage-cubes at the corners of the lens attachment each contain a mirror redirecting light by 90 • , while a knife-edge mirror which vertically splits the view is housed in the largest cube on the left in the picture.The camera-lens combination attaches to this cube which may be slided in between the neighboring smaller cubes in order to equalize the optical path lengths for the two perspectives.
The high-speed camera is operated at 250 fps with a spatial resolution of approximately 19 μm/pixel, and the resulting video data contains the top-view and side-view on the scene captured on the left and right half of the video frames, respectively.Use of the lens attachment with a conventional, i.e. a non-telecentric lens, such as in the present case, results in a significant overlap in the center of the captured images for the case of diffuse backlight illumination.Depending on the actual lens magnification settings, the overlap may obscure a huge part of the frame width when using diffuse backlight illumination for both the top and the side-view.In order to reduce the overlap, the side-view on the ice in the pool is illuminated using a collimated light source while diffuse backlighting is used for the top-view.That combination does not entirely eliminate the overlap in the center of the image, but already drastically reduces it.Due to the collimated illumination of the side-view it is relatively more the top-view overlaying on the side-view compared to the other way around.However, as shown in Fig. 2 b), the overlap concerns only a small vertical stripe in the center of the frame while the vast part of the frame is unaffected by the overlap and may serve for measurements.
Shrinkage of the droplet and the growth of the ice inside the pool are measured from the high-speed video data.The experiments are performed for a varying initial droplet size while the initial pool temperature is always at the melting temperature of water.While the constant pool temperature significantly reduces the parameter space, it also drastically increases the outcome and controllability of the experiments.It actually enables to measure the heat transferred at the pool meniscus during droplet shrinkage based on the amount of ice present after the entire pool is at the melting temperature again.Moreover, it also prevents convective currents inside the pool which would be present in the case of a warmer pool.Therefore, the draw-back from a reduced parameter space is actually fully compensated through a better control and an increased outcome of the experiments.
The measurement of the droplet size and ice volume are performed using a video post-processing algorithm implemented in a commercial software package (Matlab, The Mathworks).Apart from temporary missdetections of the droplet due to condensate clouds forming around it, the measurement of the droplet size is actually straight-forward.The droplet size is represented as the radius  of a circle being areaequivalent to the projected droplet area measured in the top-view.Note that for droplets being smaller than the capillary length of the droplet liquid,  , = √ ∕() ≈ 1.1 mm, where  and  are the surface tension and density of the liquid, respectively, and  is the gravitational constant, the liquid takes the shape of a sphere and its volume scales as  ∼  3 .However, for increasing droplet size,  >  , , the liquid gets puddle shaped.As a consequence its volume rather scales as  ∼  2  and c) the pool with an ice cap in it (dashed line).Assuming axial symmetry and accounting for the initial meniscus deformation in b), the ice volume is determined from integration of the ice cap thickness profile, i.e. the vertical distance between the pool and ice-contour in c). and thus,  does not anymore represent the radius of a sphere being volume equivalent to the droplet.
Image data used for the ice volume measurement is illustrated as an example in Fig. 3.The volume of the ice cap is determined for each video frame using the contour of the undisturbed pool meniscus (Fig. 3  a)) as a reference.Also considering the initial deformation of the pool meniscus due to the droplet weight (Fig. 3 b)), the thickness of the ice cap is measured as the vertical distance between the lower contour of the ice cap (dashed line) and the contour of the pool in its undisturbed shape (solid line), as indicated in Fig. 3 c).Measured profiles of the thickness of the ice cap along the horizontal direction resulting from cooling through a small ( 0 <  , ) and a large ( 0 >  , ) nitrogen droplet are shown as an example in Fig. 4. Depending on the initial droplet size and the associated droplet shape on the pool, the resulting shape of the ice cap significantly differs.The measurements (solid line) are shown together with a fit of the data (dashed line) based on a Gaussian ansatz function and a variation of that, respectively.While a Gaussian, ℎ  () =  exp(−( − ) 2 ) +  with the fitting coefficients , ,  & , has been found to always well describe the ice contour for the case of almost spherical droplets,  0 <  , , an appropriate variation of that, ℎ  () =  exp(−( − ) 4 ) + , well describes the situation for the case of  0 >  , .Assuming axial symmetry of the ice cap, the ice vol-Fig.4. Example ice thickness profile derived from video post-processing of experimental data for varying initial droplet size of  0 <  , in a), and  0 >  , in b).Depending on the initial droplet size, the raw data measured from the high-speed videos is fitted using a Gaussian ansatz function ( 0 <  , ) or a variation of that ( 0 >  , ); the respective ansatz functions are given in the graphs.The shown data is centered around  = 0 based on the found fit function for the raw data and the shaded region is assumed to represent the projection of the axisymmetric ice cap.Also accounting for the initial deformation of the pool meniscus due to the droplet weight, the ice volume is finally obtained from integration of the fit function in polar coordinates.ume is determined from integration of the found fit function in polar coordinates.Finally, the determined ice volume is corrected with the volume corresponding to the initial meniscus deformation due to the droplet weight as shown in Fig. 3 b).

Numerical
A commercial finite-element simulation software (COMSOL Multiphysics) is used to numerically simulate droplet shrinkage on the pool.In the numerical simulations the full time-dependent problem of a Leidenfrost droplet on an isothermal liquid pool is computed assuming axial symmetry of the problem.The numerical domain comprising the different fluid regions and boundary conditions is shown in Fig. 5. Its size corresponds to at least 20 ×  0 in both the r-and z-direction for all computations, representing an infinite surrounding for the droplet.Noslip and zero-flux boundary conditions are imposed on all outer domain boundaries which are not considered as a symmetry plane.The temperature of the pool region 3 and droplet region 1 are each assumed to be constant and thus, heat transfer and solidification are neglected in the liquid regions.However, all remaining physical mechanisms being relevant in the given situation, i.e. conservation of energy, momentum and mass, are accounted for in all involved phases.
The interfaces of the droplet and the pool are discretized using a moving-mesh, in which the phase interfaces are modeled as a geometrical surface separating the different fluid regions.The local interface velocity is tracked by taking into account both momentum and mass Fig. 5. Schematic of the axisymmetric numerical domain used for the simulation of droplet shrinkage on a liquid pool in the commercial simulation software COMSOL Multiphysics, including the boundary conditions applied around the wedge-shaped domain, and the fluid regions corresponding to the nitrogen droplet 1 , the gaseous ambient 2 and the water pool 3 .Any boundary conditions and constraints valid at the phase interfaces inside the numerical domain, e.g. the respective interface temperatures, are imposed through Lagrange multipliers.
transport at the interface in order to determine the change of the mesh nodes location.The numerical domain is re-meshed when the mesh becomes too skewed due to significant interface deformation.The meshresolution is approximately 60 μm in the vicinity of the interfaces and gradually decreases towards the far field.
Constraints for physical quantities at the phase interfaces inside the numerical domain, e.g. the respective interface temperatures, are imposed through Lagrange multipliers.Droplet evaporation is considered by imposing a mass flux along the droplet surface in correspondence to the obtained heat flux at the interface.Momentum and energy transport are basically one-way coupled by accounting for advective energy transport.However, also the temperature dependence of the physical properties of all involved fluids is accounted for via polynomial expressions derived based on the properties library of the simulation software.
Since the fields of all physical quantities are coupled in the computational model, accurate initial conditions are required to replicate a most physical situation upon initialization, which is also required to ensure convergence of the numerical solution procedure.Appropriate initial conditions for the interface shapes and fields of the physical quantities are determined using a simpler version of the model.All flow velocities are initialized as 0 and the temperatures are initialized as   for region 1 , and   for region 2 and 3 .The flow is assumed to be incompressible and the local evaporative mass flux at the droplet bottom is computed as ṁ = − λ Δ ∕(    ), where λ is the average of the thermal conductivity of nitrogen evaluated at the temperatures of the droplet and the pool, Δ is the fixed temperature difference between the pool and the droplet, and   and   are the local height of the vapor layer and the latent heat of evaporation of the droplet liquid, respectively.In this way, the coupling between the flow and temperature fields in the vapor layer is significantly weakened, resulting in a more stable and converging simulation even for poor initial conditions.Simulating the situation for a physical time of  = 0.25 s results in a deformed steady-state geometry of the interfaces and certain corresponding flow, pressure and temperature fields, which are used as the initial conditions for the fully coupled computations, finally ensuring stable and converging computations.
In addition to full simulations of droplet shrinkage, steady-state simulations only considering conductive heat transfer in the gaseous surrounding are performed for the evaluation of the theoretical modeling of heat transfer at the droplet surface (see Fig. 9).The droplet and pool interface shapes used for these simulations are extracted from the full shrinkage simulations, and similar as in the theoretical model, the interfaces are assumed isothermal.The dimensions of the numerical domain are at least 100 × 100 times the droplet radius, and the pool temperature is imposed on all external boundaries.Similar as for the shrinkage simulation, the properties library of the simulation software is used to account for the temperature dependence of the physical properties of the gaseous nitrogen ambient.

Theoretical modeling
The theoretical modeling in the present work is based on a theoretical model for the steady-state shapes of a Leidenfrost droplet and the pool below, presented by Maquet et al. (2016) [21].The essential of that model is briefly described in the following in order to set the context required for all further extensions of the model, which are separately described afterwards.
The main purpose of the model developed in [21] is the prediction of the steady-state shapes of a Leidenfrost droplet, ℎ(), and the pool below, (), both assumed axisymmetric.A schematic of the assumed geometry is shown in Fig. 6.In the model any heat flow inside the pool and the droplet are neglected.Therefore, the pool is assumed to be homogeneous at temperature   , while the droplet temperature,   , is assumed as the saturation temperature at ambient pressure of the droplet liquid.Heat transfer between the pool and the droplet is assumed to be purely conductive and to take place only in the -direction.The evaporative flux of the droplet, which actually feeds the vapor flow in the bottom part of the droplet, is determined from the resulting heat flux along the droplet interface.Any flow in the droplet and the pool are neglected, and the shapes of the droplet and the pool meniscus are determined based on an equilibrium between hydrostatic and Laplace Fig. 7. Comparison of the theoretically predicted and numerically obtained steady-state shape of nitrogen droplets with varying size on a water pool at 273.15 K.The material properties used for the theoretical predictions are summarized in Table 1.The numerical results are obtained from simulations of droplet shrinkage performed using the commercial simulation software COM-SOL Multiphysics.Also accounting for their temperature dependence, the material properties used in the simulations are retrieved from the material properties library of the simulation software.

Table 1
Physical properties and conditions used for the theoretical predictions of the droplet and pool contour shown in Fig. 7.The properties of the droplet and the pool liquid are each evaluated at their respective temperature, while the properties of the nitrogen vapor are assumed as the mean of their respective values at   = 77.4K and   = 273.15K.All data is taken from [25,26].pressure along the curved interfaces.However, for this equilibrium the effect of the flow between the droplet and the pool is also accounted for.
For the analytical description of the interface shapes, the droplet and the pool are both split into two different regions, as shown in Fig. 6, from which one is assumed to be unaffected by the interaction and resulting flow between the droplet and the pool while the other region is assumed to be affected.Both the top part of the droplet and the outer part of the pool are assumed to be unaffected by the vapor flow between the droplet and the pool.Therefore, for these parts only the equilibrium between hydrostatic and capillary forces determines the interface shapes.For the bottom part of the droplet and the inner part of the pool, the pressure of the vapor flow is also taken into account to affect the interface shapes.Since the height of the cushioning vapor layer below the droplet is relatively small compared to the size of the droplet, the lubrication approximation is used to describe the vapor flow in between the droplet and the pool.Finally, the solutions for the inner and outer part of the pool, and the top and bottom part of the droplet, are respectively patched together at the patching radius  * .The value of this patching radius is chosen such that ℎ ′ ( * ) = 2, and it has been aposteriori verified that the exact choice of  * around the chosen value does not significantly influence the final interface shapes.All properties of the droplet and the pool are evaluated at   and   , respectively.While the vapor properties are evaluated at the respective mean temperature between the droplet and the pool in [21], in the present work, the mean of a property respectively evaluated at   and   is assumed as its average value.
The mathematical description of the problem results in a set of three coupled differential equations which is numerically solved with the appropriate boundary conditions (symmetry at  = 0, first order continuity of interface contours at  * ) using the shooting method.Theoretical predictions for the interface shapes of a nitrogen droplet and the water pool at   = 273.15K are shown as an example for varying droplet size in Fig. 7.The material properties used for the theoretical predictions are summarized in Table 1.Since experimental data for the validation of the predicted droplet and pool shape is not available from the present experiments, the theoretical predictions are compared with the interface shapes obtained from the numerical simulations of droplet shrinkage, performed as described in section 2.2.As shown in the figure, the theoretical predictions for the interface contours are in perfect agreement with the numerical results regardless of the droplet size.Both the chosen patching point and the assumption of a lubrication flow below the droplet appear to be appropriate for an accurate prediction of the steady-state interface contours; assuming the numerical simulations may serve as an accurate reference.

Present extensions of the existing model
As shown before, the theoretical model from Maquet et al. ( 2016) [21] is capable of well predicting the steady-state shape of a Leidenfrost droplet above a liquid pool, only accounting for the heat and mass flux at the bottom of the droplet.For the steady-state shape of such Leidenfrost droplets, heat and mass transfer is mostly relevant in the bottom region of a droplet, and the effect of heat and mass transfer in the top region on the droplet shape can be neglected.However, the latter may be significant for the actual droplet shrinkage rate especially if the temperature difference between the droplet liquid and the ambient is large such as in the present situation.Therefore, the original model is extended to also account for heat and mass transfer at the top part of a droplet in order to allow an accurate prediction of droplet shrinkage.Note that in that context, top and bottom of the droplet refers to the parts of a droplet split by its equator; compared to the theoretical model for the interface shapes, where top and bottom are distinguished based on the -position of the lower droplet contour at the patching radius  * .Finally, by accounting for the heat consumed for the warm-up of the nitrogen gas in the vapor layer between the droplet and the pool, also the description of heat transfer at the pool meniscus is refined.

Heat transfer at the droplet
At the droplet bottom, i.e. in the vapor layer below the droplet, radiative and conductive heat transfer between the droplet and the pool compare as where   = 5.670 × 10 −8 W/(m 2 K 4 ) is the Stefan-Boltzman constant, and   = 77.4K and   = 273.15K.The thermal conductivity of the vapor, λ = 15.59 mW/(m K), corresponds to the mean of the thermal conductivity of nitrogen evaluated at   and   , and the vapor layer thickness has been assumed as   ≈ 0.1 mm.According to Eq. ( 1), radiation between the droplet and the pool is insignificant compared to heat conduction across the vapor layer, which actually turns out as the major heat transfer mechanism in the vapor layer.A comparison of radiative and conductive heat transfer at the top part of a droplet with  = 1 mm yields where the ambient far-field temperature is assumed similar to the pool temperature,  ∞ =   .In contrast to the heat transfer situation at the bottom of the droplet, radiative heat transfer is more important at the droplet top when compared to conductive heat transfer in that region.However, since the heat transfer at the droplet top is presumably small compared to the heat transfer at the droplet bottom, radiation at the droplet top has a negligibly small contribution to the overall heat transfer to the droplet.Consequently, radiation is neglected not only for the Fig. 8. Theoretically predicted upper droplet shape for varying droplet size compared with the shape of a spherical cap such as basically assumed for the estimation of the heat transfer from the ambient to the droplet top using Eq. ( 3).
droplet bottom but also for the top part of a droplet.Moreover, heat transfer between the ambient and the droplet is assumed to induce only little convection to the gaseous surrounding and thus, conductive heat transfer between the droplet and its surrounding is assumed as the major heat transfer mechanism also at the top part of the droplet.
Assuming the ambient as pure gaseous nitrogen and its thermal conductivity as linear dependent on temperature, the conductive heat flux from an infinite surrounding at the far-field temperature,  ∞ , to a spherical droplet with radius  is found as Since the droplet size considered in the present work is well in the order or even below the capillary length of the droplet liquid, the shape of the droplet top can be well approximated as a spherical cap, as shown for varying droplet size in Fig. 8.While the droplet contour is perfectly spherical for rather small droplets, its deviation from a spherical shape slightly increases for increasing droplet size.In order to account for these deviations, the expression for the heat flux at the droplet surface is refined to read where  = 1∕ 1 + 1∕ 2 denotes the local curvature of the interface with its principal radii of curvature,  1 and  2 .Compared to Eq. ( 3), deviations from a constant curvature are accounted for and consequently, the heat flux is reduced in regions of reduced curvature (center region of a droplet) and accordingly, it is increased in regions of increased curvature (equatorial region of a droplet).While the expression accounts for local effects on a non-spherical interface, for the case of a spherical droplet with  = 2∕, the expression reduces to the common solution in Eq. ( 3).The theoretical prediction for the heat flux along the droplet interface, qℎ , is shown for two different droplet sizes in Fig. 9.The theoretical data is compared with the results of steady-state numerical simulations, q , in which only conductive heat transfer is accounted for; such as in the theoretical model.In order to only compare the modeling of the heat flux at the droplet interface, the numerically obtained interface shapes of the droplet and the pool meniscus are used to calculate the heat flux distribution according to the theoretical description.The lower data branches in Fig. 9 correspond to the heat transfer at the top part of the droplets, whereas the upper branches correspond to the situation at their bottom.The numerically obtained interface curvature is used to calculate the heat flux profile at the droplet top using Eq. ( 4), and the deviations from a clear trend for the top of the large droplet at  → 0 do not represent a physical meaning, but rather are numerical artefacts from numerically calculating the curvature for  → 0.
For the small almost spherical droplet, the vapor layer thickness,   , is at its minimum at  = 0 and increases monotonously for increasing .Accordingly, the heat flux at the droplet bottom, scaling as q ∼ 1∕  , is maximized at  = 0 and decreases for increasing .In contrast, the droplet bottom gets flattened or even of concave shape for increasing droplet size resulting in a pronounced annular neck region of minimum vapor layer thickness at 0 <   < , as shown in Fig. 7 and also reported in literature [27,28,18,21].As a consequence, the heat flux is maximized at   and decreases towards both de-and increasing .Compared to the droplet bottom, the heat flux at the droplet top is significantly smaller and less dependent on  for both the small and the large droplet.According to the numerical simulations, the heat flux at the top increases with increasing  for both droplet sizes which is qualitatively well captured in the theoretical prediction for the case of the large droplet.However, for the small droplet with an almost hemispherical apex the theoretically predicted heat flux, scaling as q ∼  ≈ 1∕, is approximately constant.As a consequence, the theoretically predicted heat flux is continuous at the equator of the small droplet, compared to a certain deviation associated with the predictions of the heat flux at the equator of the large droplet.
While the theoretical predictions qualitatively well resemble the numerical results, the heat flux is underpredicted in the model along the entire droplet surface for both droplet sizes.The relative deviation between the theoretical predictions and numerical results for the heat flux to the droplet, Δ q * = 2( q − qℎ )∕( q + qℎ ), as a function of the nondimensional radius, r = ∕, is separately shown for the droplet top and bottom in Fig. 9 b).At the bottom of the large droplet, the theoretical and numerical predictions are in very good agreement, associated with deviations well below 10% for relative radii up to r ≈ 0.9.The deviation only slightly varies for r < 0.5 while it significantly increases with increasing radial position up to approximately 23% at r = 1.In contrast, for the small droplet the deviation is significantly larger and moreover, it continuously increases for r > 0. It is below 10% only up to r ≈ 0.55 and approaches its maximum of approximately 48% at r = 1.
Compared to the droplet bottom, the data generally agrees less for the top part of the droplets.At the droplet top the deviation between the theoretical predictions and the numerical results generally decreases for smaller, i.e. more spherical droplets.However, regardless of the droplet size, it is almost constant for up to r ≈ 0.8, and slightly increases only for larger r approaching the droplet equator.For the large droplet, the basic deviation is approximately 40% going up to 60% at r = 1, while it is approximately 25% ranging up to 40% at r = 1 in the case of the small droplet.Although the droplet top well resembles the shape of a spherical cap as shown in Fig. 8, the theoretical predictions significantly deviate from the numerical results, which is presumably due to the (indirect) effect of the pool on the actual heat transfer at the droplet top which is not accounted for in the theoretical model.However, note that the relatively more pronounced deviation observed for the droplet top is also relatively less relevant for the entire situation, since the heat flow at the droplet top contributes significantly less to the total heat flow to a droplet than the heat flow at the droplet bottom.

Heat transfer at the pool
The total heat transfer at the pool meniscus can be described as split up into two contributions namely conductive heat transfer for droplet evaporation at the droplet bottom, Q , and convective heat transfer for the warm-up of the initially cold vapor leaving the gap between the pool meniscus and the droplet, Q .The conductive heat flux at the pool meniscus equals the flux at the droplet bottom, which is already described in [21] as q = λ Δ ∕  , with Δ = (  −   ).
In addition to the heat required for droplet evaporation, heat is also consumed for the warm-up of the vapor forming at the droplet bottom.Evaporation results in vapor initially being at the droplet temperature.However, due to a certain warm-up through the pool, the vapor leaves the region between the pool and the droplet with a temperature above the droplet temperature, T .Therefore, the heat flow consumed for the warm-up of the vapor flow from   to its increased averaged temperature T can be expressed as where ṁ and c, are the mass flow of evaporating vapor at the droplet bottom, and its average heat capacity, respectively.Accounting for the temperature dependence of the heat capacity of the nitrogen vapor, the average increase of the specific sensible heat of the vapor flow,  * , = c, ( T −   ), is determined as where  () is the temperature profile inside the vapor layer, and  , ( ) is the temperature dependent heat capacity as obtained from [25], which is fitted using a rational expression in order to allow solving the integral analytically.Note that for the present situation with constant temperatures of the droplet and the pool, also the average increase of the specific sensible heat of the vapor is constant as  * , = 1.22 × 10 5 J/kg.Using the relation Q = ṁ   where   is the latent heat of evaporation of the droplet liquid, Eq. ( 5) can be rewritten as Finally, the total heat flow extracted from the pool can be expressed as where    =  * , +   represents the effective latent heat combining both the latent heat of evaporation of the droplet liquid and the specific sensible heat associated with the warm-up of its vapor.For the present situation with a nitrogen droplet on a water pool at 273.15 K, the effective latent heat is found as    ∕  ≈ 1.61 [25].Accordingly, the effective cooling from a droplet is approximately 61% higher compared to the heat consumed solely for droplet evaporation.

Time discretization
Compared to the typical evaporation time of Leidenfrost droplets in the current situation, the thermal and viscous relaxation times are typically relatively short and thus, a quasi-stationary approximation for the shape evolution of the interfaces is basically sufficient for modeling of droplet shrinkage over time.Accordingly, the quasi-steady shape model is used to determine the interface shapes for a given droplet size, while the dynamics of the shape evolution are assumed to not affect any other involved processes.
Using the total heat flow at the droplet surface, Q = Q + Q , the temporal evolution of the droplet volume,  , is iteratively calculated as where Δ denotes the numerical time step, and Q is determined for every time step using the current shape of the droplet and pool interface as described above.
Finally, for every time step the solution procedure starts with 1.) the prediction of the interface geometries associated with the current droplet volume, followed by 2.) the calculation of the total heat flow and evaporative volume flux at the droplet surface, and ends with 3.) the prediction of the droplet volume in the next time step according to Eq. ( 9).For each time step, the heat flow at the pool meniscus is determined using Eq. ( 8).The time step is chosen as Δ = 0.05 s for all theoretical predictions of droplet shrinkage, resulting in an insignificant effect of the temporal discretization on the predictions.
Similar as in the model from Maquet et al., (2016) [21], heat transfer in the pool is neglected in the present model and the pool temperature is assumed to remain constant at its initial temperature.Neglecting any temperature changes in the pool and in the formed ice and thus, assuming that all heat transfer at the pool meniscus only results in ice formation in the pool, the temporal evolution of the ice volume can be estimated as where   represents the latent heat of solidification of the pool liquid.  () denotes the heat extracted from the pool until time , which is obtained from numerical integration of the total heat flow at the pool meniscus as predicted for each time step.

Results
An example image sequence of droplet shrinkage and ice growth is shown in Fig. 10.For the shown sequence the original video data, which contains both perspectives in the same video frame, has been cropped to the respective region of interest for both perspectives.Two unprocessed high-speed videos showing the situation for a small and a large droplet as captured in the experiments are found as an example in the supplementary material to the present work.Time  = 0 in Fig. 10 refers to the moment when the droplet makes first contact with the pool meniscus.Starting with an initial droplet radius of  0 ≈ 0.86 mm, droplet evaporation due to the heat originating from the ambient and the pool leads to a continuous decrease of the droplet size until the droplet is completely evaporated at  ≈ 6.5 s.The heat dissipated from the pool for droplet evaporation and vapor warm-up results in the growth of an ice cap below the droplet.Since the pool liquid is initially at its melting temperature, the size of the ice cap remains almost constant after completed droplet evaporation and thermal equilibration of the ice cap with the pool, and finally it is representative for the heat extracted from the pool.
The experiment has been repeated for an initial droplet size varying between  0 ≈ 0.6 mm and  0 ≈ 1.75 mm, and a pool initially always being at its melting point.Subdivided into the processes taking place on top and below of the pool meniscus, the results for the droplet dynamics and ice formation are separately discussed in the following sections.

Droplet dynamics
Primarily focusing on the processes taking place on top of the pool meniscus, the droplet dynamics and involved phenomena are separately discussed in the following.After a scaling analysis for the droplet size evolution during shrinkage, the experimental, numerical and theoretical results for droplet shrinkage are quantitatively compared.The effect of droplet shape oscillations on the experimental measurements is demonstrated, and the theoretical and numerical predictions for the heat flow to the droplet top and bottom are examined for varying droplet size.Finally, the effect of the initial droplet size on droplet shrinkage and the role of the ice cap in the present situation are elucidated.

Scaling analysis for droplet shrinkage
The theoretical prediction for the temporal evolution of the radius of a nitrogen droplet with  0 = 2.5 mm as obtained from the theoretical model for droplet shrinkage is shown as a solid line in Fig. 11.The dotted line in the figure represents a linear fit of the data for 10 s ≲  ≲ 15 s.As indicated through differently colored regions in the figure, the life time of the droplet can be subdivided into three different shrinkage regimes relating to the actual droplet size.The associated interface shapes for varying droplet size are shown as an example for each regime.12) -( 14).Theoretically predicted interface shapes being in relative scale to each other are shown as an example for the different regimes.The dotted line represents a linear fit of the data in regime II, i.e. for 10 s ≲  ≲ 15 s.
For the case of conductive heat transfer below the droplet being the major contribution to the total heat flow, the dominating behavior in the different shrinkage regimes can be estimated based on the associated droplet shape and the major effects on the heat flow at the droplet bottom.Generally, the evaporative volume flow resulting from the heat flow at the droplet bottom scales linearly with the relevant bottom area of the droplet, , and the reciprocal of the vapor layer thickness   as from which the shrinkage regimes may be derived as follows.
Regime I For the case of small spherical droplets with  ≪  , , droplets take a fully spherical shape.As a consequence, the distance between a droplet and the pool scales with the droplet radius,   ∼ , and the bottom surface area scales as  ∼  2 .Therefore, the total heat flow at the bottom of a small droplet scales as Q ∼ , which together with  ∼  3 results in a scaling law for the shrinkage of small droplets as where  represents the time constant of the process.
Regime II In the case of medium sized droplets with  ≈  , , the major part of the droplet can still be well approximated as spherical and thus,  ∼  3 , while the droplet bottom becomes flattened and scales as  ∼  2 .The vapor layer thickness can be assumed to be invariant to the droplet size and thus, the heat flow at the bottom scales as Q ∼  2 , which results in a scaling law for the droplet radius as Regime III Finally, for large droplets with  ≫  , , the liquid gets flattened and takes the shape of a puddle whose height is invariant against its extent on the surface, and thus,  ∼  2 .Similar to medium sized droplets, the vapor layer thickness is rather independent of  such that the heat flow at the liquid bottom scales as Q ∼  2 , from which a scaling law for the radius of large droplets is found as Note that the thresholds separating the regimes for the evolution of the droplet radius in Fig. 11 are chosen arbitrarily.Nevertheless, the shrinkage regimes corresponding to Eqs. ( 12) -( 14) can be well identified in the theoretical prediction for droplet shrinkage compared to the linear fit of the data for the medium sized droplet.

Quantitative results for droplet shrinkage
A comparison of the experimentally, theoretically and numerically obtained temporal evolution of the droplet radius for varying initial droplet radius,  0 , is shown in Fig. 12.Time  = 0 corresponds to the moment when the droplet makes first contact with the pool liquid.Since the droplet generator may obscure the view on the droplet during its deposition (see Fig. 10), data for early times after first contact is not available for all experiments, while  = 0 can be still well determined from the captured side-view.In the case of an obscured top-view for early times, the initial droplet radius is determined from extrapolation of the existing measurements to  = 0.
As shown in the figure, the theoretical predictions and numerical results for the droplet shrinkage are in excellent agreement for the small droplet and moreover, both are in almost perfect agreement with the experimental data.The evolution of the droplet radius is perfectly predicted theoretically and numerically for almost the entire lifetime of the droplet.Only during the last phase of the droplet life time, the theoretical and numerical predictions slightly overpredict the shrinkage rate compared to the experimental observations.However, the effect of this overprediction is negligibly small and consequently, compared to the experimental data the theoretical and numerical predictions for the droplet evaporation time until  = 0.25 mm are associated with a negligible error of only 3.1% and 1.0%, respectively.
For the medium sized droplet, the droplet shrinkage rate is slightly overpredicted using the theoretical model compared to the numerical results.While the latter still show a good agreement with the experimental data for almost up to 75% of the droplet life time, the theoretical data overpredicts the shrinkage rate significantly earlier.Only for later times also the numerical results show a pronounced overprediction of the droplet shrinkage rate.However, similar to the case of the small droplet, the effect of the overprediction is comparably small and results in a relatively small error of only 2.6% and 6.2% for the numerical and theoretical predictions of the evaporation time until  = 0.25 mm, respectively.Finally, also the numerical results for the shrinkage of the large droplet are in very good agreement with the experimental data for the majority of the droplet life time, i.e. for even up to 80% of the life time.For later times, the numerical results are associated with a significant overprediction of the shrinkage rate observed in the experiments.Compared to the case of the medium sized droplet, the overprediction of the shrinkage rate by the theoretical model is even more pronounced.Nevertheless, the agreement in terms of the evaporation time until  = 0.25 mm is still relatively good for both the theoretical and numerical predictions.While it amounts 7.3% for the theoretical predictions, it is still comparably small for the numerical results and values to only 2.6%.
Concluding, although the overprediction of the droplet shrinkage rate in the theoretical and numerical results significantly increases with increasing droplet size, the numerical and theoretical results show a Fig. 13.Image sequence of the largest droplet in Fig. 12 showing a full period of a harmonic shape oscillation in mode n=2.The shown oscillation is observed at  ≈ 3.2 s after first contact between the droplet and the pool, while time  * = 0 refers to the arbitrarily chosen begining of the oscillation period.Arrows in the images indicate the instantaneous local direction of motion of the droplet contour.Droplet shape oscillations cause significant fluctuations of the measured droplet radius such as shown in Fig. 12, but are assumed to be without a significant effect on the average evolution of droplet shrinkage.good overall agreement with the experimental data for all droplet sizes in the present study.In particular, the numerical results well agree with the experimental data for all situations and allow accurate prediction of droplet shrinkage.Nevertheless, taking into account the various physical mechanisms involved in the situation, also the theoretical model enables a comparably accurate prediction of droplet shrinkage.

Droplet shape oscillations
The pronounced fluctuations of the experimental data for the droplet radius shown in Fig. 12 can be attributed to shape oscillations of the droplet, which are most notably observed for the largest droplet.An image sequence of the largest droplet in Fig. 12 during a full period of a harmonic shape oscillation is shown as an example in Fig. 13.Shape oscillations during droplet shrinkage can also be observed in movie 2 supplied with the present work as supplementary material.
Harmonic shape oscillations have already been reported for Leidenfrost droplets on a curved solid substrate [29][30][31][32][33][34][35], or the curved meniscus of a liquid pool [20].However, for the present case, it is worth to note that although the droplet is deposited on an initially liquid pool, the pool liquid below the droplet is not anymore liquid when the shape oscillations are observed.Instead, the nitrogen droplet hovers above a solid ice cap of approximately the same diameter as the maximum extent of the oscillating droplet.The ice cap starts to form at early times after droplet deposition and the initially curved shape of the pool meniscus is actually preserved through freezing.Therefore, the present situation is actually well comparable to the shape oscillations reported for Leidenfrost droplets on a curved solid substrate.Consequently, also the dimensionless oscillation frequency as observed in the present experiments,  ∕ √ ∕ , ≈ 0.25, well compares to the frequency of 0.2 as reported for a nitrogen Leidenfrost droplet on a curved solid substrate [33].However, in contrast to oscillation modes solely with  > 2 reported for such droplets in [20,33], harmonic oscillations with  > 2 are rather an uncommon exception in the present experiments and the most dominating oscillations are observed in mode  = 2. Nevertheless, also the relation between the observed frequency and the oscillation mode as a function of the droplet size well agrees with theoretical considerations in [29,20].
While the observed shape oscillations generally add further dynamics to the situation and cause significant fluctuations of the measured droplet radius as shown in Fig. 12, their effect on the physics controlling droplet shrinkage is presumably negligibly small, as also implied by the good agreement of the theoretical and numerical data with the experimental results.

Heat to the droplet
The theoretical predictions for the heat flow to the droplet top and bottom during droplet shrinkage are shown in Fig. 14 for varying droplet radius.The data is compared to the heat flow obtained from the full numerical simulations of droplet shrinkage.In addition to the absolute data for the heat flows, the relative contribution of the top heat flow to the total heat flow to the droplet, Q ∕ Q , is also shown in the figure.Note that in contrast to the comparison of the heat flux at the droplet surface shown in Fig. 9, in the present case, also convective heat transfer in the ambient of the droplet is accounted for in the numerical simulations.Moreover, in the present case the theoretical prediction of the interface shape is used to determine the heat fluxes and resulting heat flow at the droplet surface within the theoretical model.
As shown in Fig. 14, the theoretical predictions and numerical results qualitatively well agree for both the top and the bottom of a droplet.The heat flow at the droplet bottom is always larger than at the droplet top and thus, it generally dominates the heat transfer situation for all present droplet radii.Both contributions increase for increasing droplet size, almost following a power-law dependency for the entire range of droplet sizes.Although the theoretical predictions for the heat flow at the droplet qualitatively well resemble the numerical results, they quantitatively slightly differ.As found from fitting a power-law ansatz function to the data, both the theoretical and numerical results predict a scaling of Q ∼  1.57 for the bottom part of the droplet.In contrast, the theoretically and numerically predicted scaling for the droplet top significantly differs as Q,ℎ ∼  0.92 and Q, ∼  1.39 , respectively.Consequently, also the theoretical and numerical results for the relative contribution of the heat flow at the droplet top to the total heat flow to the droplet, Q ∕ Q , significantly differ.Compared to the numerical results, the contribution is theoretically predicted to be significantly more dependent on the droplet size.Although both give a good indication about the relative effect of the heat flow to the top of the droplet, the numerical data is actually assumed to more accurately reflect the actual heat transfer to the droplet.
While the total heat flow at the droplet top contributes even more than 30% to the total heat flow for the case of the smallest spherical droplets, its relative effect continuously decreases for increasing droplet size, i.e. for a flattening of the droplet.Nevertheless, also for large droplets, the relative contribution of the heat transfer at the droplet top remains significant and amounts to approximately 10% for  = 1.7 mm, which finally reveals that heat transfer from the ambient to a Leidenfrost droplet can not generally be neglected.In particular for the case of huge temperature differences between the ambient and the droplet, such as in the present situation involving a cryogenic Leiden-Fig.15.Experimental data for the droplet radius evolution for varying initial droplet radius.Compared to the representation in Fig. 12, the data is shown on a shifted time axis,  ′ =  −  0.4 , where  0.4 corresponds to the time after droplet deposition when an individual droplet's radius is  = 0.4 mm.frost droplet, disregard of the heat flow at the droplet top may cause significant under-estimation of the total heat flow to a droplet, and the resulting droplet shrinkage rate.

Effect of the initial droplet size on shrinkage
The data for the temporal evolution of the droplet radius for varying initial droplet size of all performed experiments is shown in Fig. 15.The data is shown on a shifted time axis,  ′ =  −  0.4 , where  0.4 corresponds to the time when an individual droplet's radius is  = 0.4 mm after its first contact with the pool.As a consequence, all droplet shrinkage curves intersect at  ′ = 0 in order to illustrate the effect of the initial droplet size on droplet shrinkage.
After a droplet is placed on the pool and ice forms in the pool, a temperature gradient develops in the ice cap and the temperature of the top of the ice cap falls below the initial pool temperature.Intuitively, this cool down of the ice cap below the droplet could be expected to significantly affect droplet shrinkage depending on the initial droplet size; in particular for the present situation with a droplet at 77.4 K.However, as shown in the figure, all curves almost perfectly overlap regardless of the initial droplet size.Independent of any additional pre-cooling in the case of an initially larger droplet, the instantaneous shrinkage rate of a droplet primarily depends on the current droplet size; and counterintuitively it is only insignificantly affected by the initial droplet size.Note that the situation may change for other conditions such as a pool initially being at a different temperature.However, for the present conditions the initial droplet size has almost no effect on the droplet shrinkage rate for a given droplet size, which can be explained with the insignificant cool-down of the ice cap surface during droplet evaporation.
Example numerical results for the temporal evolution of the surface temperature of a semi-infinite slab of water are shown in Fig. 16.The water is initially at its melting temperature and a constant heat flux, q = 20 kW/m 2 , is applied to its surface at  = 0.The applied heat flux corresponds to a typical heat flux at the pool meniscus below a droplet as obtained from the numerical simulations, in order to replicate the situation inside the pool.The resulting temporal evolution of the surface temperature is compared for the cases of pure heat conduction with and without phase change accounted for in the water.As shown in the figure, the presence of ice significantly reduces the cool-down of the surface compared to the case without phase-change in the pool.If ice is present, all heat transfer is restricted to take place inside the ice and the release of latent heat of fusion from the slowly moving solid-liquid interface drastically lowers the surface cool-down compared to the case of pure heat conduction without phase change.After 10 s which is well comparable to typical evaporation times in the present experiments, the ice surface is cooled down by only approximately 7 K which is very small compared to the typical temperature difference in the problem, Fig. 16.Numerical results for the temporal evolution of the surface temperature of a semi-infinite slab of water initially being at 273.15 K.At  = 0, a constant heat flux, q = 20 kW/m 2 , is applied to the surface, finally reflecting the situation in the pool below a deposited droplet.The evolution of the surface temperature due to heat conduction in the water is compared for the situation with,  , , and without phase change and the resulting ice accounted for,  , .
Δ = 196 K.In contrast, the surface cools down by even almost 50 K for the case of pure heat conduction, which would presumably be indeed significant for the situation.However, due to ice formation in the present experiments, droplet shrinkage is almost invariant with respect to the initial droplet size and consequently, both the theoretical and numerical predictions well agree with the experimental results, even though they involve the assumption of a constant pool temperature.Note that the present simulations refer to 1D heat conduction in cartesian coordinates.Therefore, in the real situation the curved solidliquid interface of the ice cap will even more suppress the cool down of the top of the ice cap.However, for other conditions such as an initially increased pool temperature, the theoretical and numerical model are probably less accurate to predict droplet shrinkage than in the present situation.The initial cool down of the pool liquid below the droplet may be too pronounced to justify the assumption of a constant pool temperature.Nevertheless, the assumption of a pool temperature at the melting temperature is presumably reasonable to predict the shrinkage behavior for times after ice has formed below the droplet.

Ice formation
The temporal evolution of the ice volume forming inside the pool as experimentally measured and theoretically estimated using Eq. ( 10) is shown for varying initial droplet size in Fig. 17.Several problems may accompany the ice volume measurement at the very begining of an experiment such as a pool meniscus moving after droplet deposition, or a delayed onset of freezing which may result in liquid supercooling and its dendritic freezing.Consequently, the ice volume can not be accurately measured during a certain time after droplet deposition and thus, no experimental data is shown for short times after droplet deposition in Fig. 17.The black crosses in the figure correspond to the moment of completed droplet evaporation in the experiment.
For early times after droplet deposition, the experimentally measured ice growth rate is almost constant and increases with increasing droplet size.In contrast, the theoretically estimated ice growth rate is initially increased compared to the experimental data and starts decreasing immediately after droplet deposition.Since thermal gradients in the ice are neglected in the theory, the ice volume is over-predicted compared to the experimental data for most of the time.However, for late times the over-prediction of the ice volume continuously decreases due to a decreasing droplet size and thus, a decreasing heat flow at the top of the ice cap.With decreasing heat flow, also the thermal gradients in the ice cap reduce until they finally almost vanish out at the end of the life time of the droplet.As a consequence, after full evaporation of the droplet also the major ice growth is finished and thermal gradi-Fig.17.Temporal evolution of the ice volume growing inside the pool for varying initial droplet size as measured in the experiments (solid line) and theoretically predicted using Eq.(10) (dashed line).The black symbols correspond to the moment when the droplet is completely evaporated in the experiment.Although the theoretical model over-predicts the ice volume as long as the nitrogen droplet is still on the water pool, the total heat extracted from the pool for droplet evaporation, as reflected in the final ice volume, is well predicted using the model.A comparison of the theoretical predictions with the experimental results for the final ice volume depending on the initial droplet size is shown in Fig. 18.Note that in some experiments the ice cap moved sideways which prevents reliable measurement of the ice volume.In the case of significant movement of the ice, its volume measurement is disregarded and as a consequence, data is not shown for all performed experiments.As already implied by the temporal evolution of the ice volume shown in Fig. 17, the theoretical predictions for the final ice volume are in good agreement with the experimental data for all examined droplet sizes.Apart from one exception with a larger error for  0 ≈ 0.65 mm, the relative error of the theoretical predictions is well below 10% for all remaining experiments in the present study, finally again confirming the good predictive capabilities for the present situation of both the numerical simulations and the theoretical model.

Conclusion
The shrinkage of a nitrogen Leidenfrost droplet on a water pool has been examined experimentally, theoretically and numerically.For the experiments, a nitrogen droplet is deposited on a water pool and the evolution of the droplet size and ice volume in the pool is measured from high-speed video data showing droplet shrinkage and ice formation in a top-and a side-view, respectively.While the water pool is at its melting temperature at the begin of all experiments, the initial droplet radius is varied between  0 ≈ 0.6 and  0 ≈ 1.75 mm.
An existing theoretical model for the interface shape of the droplet and the pool, developed by Maquet et al. (2016) [21], has been extended to predict the temporal evolution of the droplet size and the volume of ice formed below the droplet inside the pool.While only conductive heat transfer at the droplet bottom is accounted for in the original model, in the present work the model has been extended to also account for conductive heat transfer from the ambient to the droplet top.Moreover, the heat transfer at the pool meniscus below the droplet due to the warm up of the vapor generated at the droplet bottom, which leaves the vapor layer with an increased temperature, is considered in the model.It allows to predict the heat extracted from the pool and thus, the rate of formation of ice in the pool, which is compared to the ice volume measured in the experiments.
Numerical simulations of droplet shrinkage and the involved mechanisms have been performed using the commercial finite-element simulation software COMSOL Multiphysics.The numerical results are compared to theoretical predictions, and experimental data if that is available.For both the theoretical model and the numerical simulations, heat transfer in the pool is neglected and the pool meniscus is assumed to be constant at its initial temperature.
It has been shown that droplet shrinkage in the present situation is almost invariant towards the initial droplet size and the droplet shrinkage rate mainly depends on the instantaneous, but not on the initial droplet size.Numerical simulations revealed that the presence of ice significantly reduces the cool down of the pool meniscus during droplet shrinkage compared to the hypothetical absence of ice and pure heat conduction in the pool.Compared to the initial temperature difference between the droplet and the pool, the surface temperature reduces only insignificantly, finally explaining the invariance of droplet shrinkage against the initial droplet size.
Therefore, the assumption of a constant pool temperature for the present situation with a pool initially at its melting temperature turns out reasonable and thus, both the theoretical model and numerical simulations generally well predict droplet shrinkage for all examined droplet sizes.For the smallest droplet size, both the theoretical predictions and the numerical results are in perfect agreement with the experimental data.However, for increasing droplet size the theoretical model and numerical simulations increasingly over-predict the droplet shrinkage rate.Nevertheless, both stay in reasonable agreement with the experimental results for all droplet sizes, though the numerical results show less deviation from the experimental data.
The theoretical predictions for the ice volume forming inside the pool well agree with the experimental data.Due to thermal gradients inside the ice cap, which are not considered in the theoretical estimation, the ice volume is significantly over-predicted for times in which the droplet represents a considerable heat sink for ice formation.However, with decreasing droplet size, the heat flow to the droplet reduces and thus, also the thermal gradients inside the ice reduce and finally vanish out.For times after completed droplet evaporation, i.e. with the ice cap being in thermal equilibrium with the surrounding liquid, the predicted ice volume well agrees with the experimental data.
Concluding, both the theoretical model and numerical simulations turn out to be well capable for the prediction of the shrinkage of a nitrogen Leidenfrost droplet on a liquid water pool initially at its melting temperature.The good agreement for both droplet shrinkage and ice formation confirms the accurate modeling of the major involved mechanisms and the resulting predictive capabilities for the present situation.However, due to the assumption of a pool at its melting temperature, their capabilities for the prediction of the situation with a pool initially above its melting temperature are limited.The predictions for early times after droplet deposition which may be associated with a significant cool down of the pool liquid below the droplet, might be associated with a significant error.However, for times after ice has formed below the droplet, the theoretical and computational models are expected to be as capable as in the present situation.Finally, the theoretical and computational model represent powerful tools for the prediction of sit-

Fig. 1 .
Fig. 1.Schematic side view of the double-walled transparent container used to provide a temperature controlled water pool (light blue) for the experiments.A transparent cooling fluid (dark blue) pumped around the liquid pool ensures a homogeneous temperature distribution inside the pool.Blue arrows indicate the flow direction of the cooling fluid.A droplet is placed on the pool meniscus provided in a circular cut-out of a thin kapton foil closing off the bottom of a top-insert extending down into the pool.The situation is observed from the top and the side as indicated through pictograms using a high-speed video system.(For interpretation of the colors in the figure, the reader is referred to the web version of this article.)

Fig. 2 .
Fig. 2. Model view of the lens attachment used for the experiments in a) and an unprocessed video frame captured from the scene in b).The lens-attachment is composed from opto-mechanical components (Thorlabs) and allows to capture the two perspectives on the situation using a single camera as indicated in Fig. 1.The blue and orange lines in a) and the frames in b) represent the optical paths for the two perspectives, and the corresponding regions for the top and side view in the captured frame, respectively.The camera-lens system connects to the left side of the lens attachment where the blue and orange arrows leave the largest cage-cube housing a knife edge mirror, resulting in a top-view on the left and a side-view on the right of the video frames, as shown in b).See the online version of the manuscript for reference to colors.The snapshot of the CAD assembly in a) is reprinted with permission from Thorlabs.

Fig. 3 .
Fig. 3. Example side-view images illustrating the measurement of the ice volume in the pool based on the detected contours of a) the undisturbed pool (solid line), b) the pool with only a small dimple from the droplet on top (dotted line),and c) the pool with an ice cap in it (dashed line).Assuming axial symmetry and accounting for the initial meniscus deformation in b), the ice volume is determined from integration of the ice cap thickness profile, i.e. the vertical distance between the pool and ice-contour in c).

Fig. 6 .
Fig. 6.Schematic illustration of the geometrical situation of a Leidenfrost droplet on a liquid pool, such as assumed for theoretical modeling of the problem by Maquet et al. (2016) [21].The entire situation is assumed axisymmetric, as shown in a).A detail of the scene around the vapor layer as indicated with the shaded rectangular area is shown in b).The interface contours of the droplet, ℎ(), and the pool, (), are separately determined for different solution regions, where the patching radius  * radially splits the pool contour into an outer and an inner part while the point of the lower droplet contour at  * vertically separates the droplet contour into a top and a bottom part.

Fig. 9 .
Fig. 9. Heat transfer to a Leidenfrost droplet depending on droplet size: a) Comparison of the steady-state radial profile of the theoretically predicted, qℎ , and numerically obtained, q , heat flux at the droplet interface for different droplet sizes of  ≈ 0.5 mm (gray lines) and  ≈ 1.1 mm (black lines).While the lower branches respectively correspond to the heat flux predicted for the top part of the droplets, the upper branches represent the heat flux at the droplet bottom.b) Relative deviation between the theoretical and numerical results from a), separately shown for the top and bottom of the droplets as a function of the non-dimensional radius, r = ∕.

Fig. 10 .
Fig. 10.Example image sequence showing droplet shrinkage on (top-row) and ice growth in the water pool (bottom-row) in a top-view and a side-view, respectively.Time  = 0 refers to the moment when the droplet makes first contact with the pool meniscus and the images column-wise refer to the same time.The dark region in the first two frames of the top-view represents the dispenser device used to generate and deposit the nitrogen droplet.The initial droplet radius is  0 ≈ 0.86 mm theoretically resulting in a final ice volume of   ≈ 1.4 mm 3 .

Fig. 11 .
Fig. 11.Theoretical prediction for the shrinkage of a nitrogen droplet with an initial radius of  0 = 2.5 mm.The colored regions indicate the different shrinkage regimes in which the evolution of the droplet radius scales according to Eqs. (12) -(14).Theoretically predicted interface shapes being in relative scale to each other are shown as an example for the different regimes.The dotted line represents a linear fit of the data in regime II, i.e. for 10 s ≲  ≲ 15 s.

Fig. 12 .
Fig. 12.Comparison of the theoretical, numerical and experimental results for the temporal evolution of the droplet radius, , for varying initial radii of  0 ≈ 0.65 mm,  0 ≈ 1.06 mm and  0 ≈ 1.75 mm.The shaded regions correspond to the different shrinkage regimes as illustrated in Fig. 11 and discussed in Sec.4.1.1.

Fig. 14 .
Fig. 14.Comparison of the theoretical predictions and numerical results for the heat flow to the droplet top and The data is shown for varying droplet radius and the relative contribution of the heat flow at the droplet top to the total heat flow, Q ∕ Q , is also included.Note the logarithmic axes for  and Q (left axis), and linear axis for Q ∕ Q (right axis).The colored regions in the graph correspond to the shrinkage regimes highlighted and discussed in the scope of Fig. 11.

Fig. 18 .
Fig. 18.Comparison of the theoretical predictions with the experimental results for the final volume of ice formed inside the pool depending on the initial droplet radius.