Freezing Process Simulation of Standing Water Droplets Under Different Supercooling and Contact Angles

,-,-,

(School of Naval Architecture&Ocean Engineering,Jiangsu University of Science and Technology,Zhenjiang 212003,China)

Abstract: In order to eliminate danger and to reduce the frequency of ship accidents caused by ice forming on ship decks and superstructures,it is important to have a physical understanding and the ability to predict icing.By studying the freezing stages in an icing process, the influence of different parameters on the icing process is analyzed, which provides a theoretical basis for the prediction of ship icing.In this paper, the proportion of the ice phase after the water droplet recalescence stage is calculated, the water droplet profiles at different stages are discussed, and the freezing time of water droplets at different supercooling degrees and contact angles is simulated by Fluent.The results show that both the supercooling degree and the contact angle have a great influence on the freezing time of water droplets.The decrease of supercooling degree and the increase of freezing time are approximately exponential functions, and the increase of contact angle is also exponential functions for the increase of freezing time.

Key words:ship icing;ice phase proportion;supercooling;contact angle;freezing time

0 Introduction

Ships operating in polar environments often face the problem of superstructure icing caused by marine droplets.A large amount of icing will affect the stability of offshore structures and the maneuverability of deck equipment, and even endanger the safety of personnel[1].Therefore, the polar ship icing problem has attracted the attention of scholars in recent years.Atmospheric icing and sea spray icing are the two types of marine icing in cold ocean regions and more than 80% of the icing events are caused by sea spray[2].The prediction model of ice accretion on offshore structures has been extensively studied and is largely based on empirical formulas and data statistics[3].Both air and sea spray icing involve the transmission of heat and the freezing of droplets.Moreover,since the freezing process of water droplets is extremely sensitive, any external disturbance will affect the final ice type (bright ice,frost ice and mixed ice) and physical property parameters[4].Meanwhile,the surface characteristics of materials will also have an impact on the adhesion and icing results of ice accumulation.Therefore, in order to obtain accurate prediction results of ship icing, it is necessary to consider the microscopic icing process of water droplets on the surface, and analyze the icing results of water droplets under different parameters from the microscopic perspective,so as to provide a certain theoretical basis and technical reference for future research on ship icing prediction and improvement of anti-icing/deicing related engineering applications.

In recent years,progress has been made in the simulation of water droplet icing by using phase field and numerical methods.The phase field approach, which is based on Ginzburg-Landau theory, is a simulation technique that uses differential equations to represent the diffusion, ordering potential, and mechanical driving processes of matter[5].Bai[6]used the finite difference approach based on the Wheeler phase field model to estimate how saltwater would solidify at various supercooling temperatures.Fan[7]developed a phase field model to explain the development of ice crystals and dynamics of mesoscale freezing concentration based on the average field approximation of solute concentration and the transport phenomena of latent heat, mass, and momentum.Han[8-9]developed a computational model to simulate the formation of hexagonal ice crystals from pure water.Then,based on Wheeler model,the numerical simulation of sea water ice crystal was realized by using sea water physical parameters,which provided a theoretical basis for the rewiring stage of supercooled water droplet freezing and further revealed the freezing mechanism.The phase field method is mostly used to study the nucleation and recalescence stage of the freezing process of water droplets.For the research of the freezing stage of static water droplets,the main methods adopted now include theoretical model and numerical simulation.Hindmarsh[10]further compared two kinds of spherical supercooled water droplet freezing models.After comparing the water droplet temperature change curve calculated by the model with the experimental measured value, it was found that the model with the phase interface moving from the inside to the outside was more practical.Zhang[11]adopted the equivalent specific heat capacity method based on the initial profile of water droplets.It was found that the surface contact angle would seriously affect the freezing process of water droplets.For water droplets of a given size, the freezing time increased exponentially with the increase of the wall contact angle.Tryggvason[12]combined the front tracking method and the finite difference method to numerically simulate the freezing process of a single droplet on the wall, and well reproduced the contour change of the droplet freezing process.Zhang[13]conducted a freezing experiment of 10 μL water droplets on a cold wall surface with a temperature of -16.5 ℃, capturing the evolution of the solid-liquid interface inside the droplets during the freezing process.Compared with the hydrophilic surface, the results of the hydrophobic surface are in good agreement with the experimental results.As mentioned above, when studying the microscopic freezing process of water droplets, the recalescence process is often ignored, or the recalescence phenomenon is only observed,without associating it with the freezing stage.

The main objective of this paper is to establish a model to simulate the freezing of static water droplets.The model introduces the influence of the degree of supercooling on the recalescence stage and considers the influence of the recalescence stage on the freezing process of the droplet,which affects the initial physical parameters of the main freezing stage of the droplet.Calculation of ice phase proportion by recalescence phase with phase field method according to the result after the recalescence and freeze phase droplets by physical parameters, the solidification/melting model by using FLUENT software to different degree of supercooling and contact angle of the static water ice freezing process simulation, and system analysis of the different degree of supercooling and contact angle influence on freezing stage,provide a theoretical basis for ship icing prediction and anti-icing research.

1 Frozen water drop theory

As shown in Fig.1,the frozen water droplets can be divided into five stages:liquid cooling,nucleation, recalescence, freezing and solid cooling[14].In the stage of liquid cooling and nucleation, the droplet gradually cools down on the cold surface until the temperature reaches the critical nucleation temperature, and then spontaneously growing crystal nuclei larger than the critical size are randomly formed inside the droplet.The supercooling degree of the supercooled water droplet is released during the freezing phase, and its internal temperature maintains the temperature of the ice water mixture throughout the freezing phase.The recalescence stage is the process of the growth and release of heat of the crystal nucleus after the nucleation of the water droplet in the supercooling.Meanwhile, the interface of the ice water phase gradually advances from the bottom to the top of the droplet until it is completely frozen under the continuous drive of the cold surface, which lasts a long time.The solid cooling phase is the last stage of the freezing process,when the temperature inside the droplet slowly decreases again until it is the same as the cold surface temperature.

1.1 Recalescence process calculation

During the liquid solidification process,the latent heat will be released when the water changes from liquid phase to solid phase, which will be transferred to the supercooled liquid around the solid phase through the solid-liquid interface.In the process of liquid solidification, the generation of latent heat and the cooling of liquid surface to the surrounding environment compete with each other,and the temperature of liquid depends on the results of competition between the two processes.After the supercooled liquid begins to solidify, the solidification rate is usually fast.Therefore,the heat release rate from the solid-liquid interface to the supercooled liquid is much faster than the heat dissipation rate from the liquid surface to the surrounding environment, which will lead to the increase of the liquid temperature , namely the recalescence phenomenon of the liquid.In this paper,the phase field method is used to calculate the recalescence stage.

The phase field equation is:

whereΔis dimensionless subcooling degree,Δ=cΔT/L,cis sensible heat,Lis latent heat per unit volume,mis phase field mobility,ais interfacial free energy.

The temperature field equation[15]is:

where,p(ϕ)=ϕ3(10 - 15ϕ+ 6ϕ2),uis dimensionless temperature,u=(T-TM)/(TM-T0).

This paper focuses on the solidification of water droplets in the polar region.Considering that most of the polar regions are iced seawater,seawater is regarded as a binary solution.In the process of seawater solidification, since the solubility of solute salt in water is greater than that in ice, the solute will be enriched at the front of the solid-liquid interface.As the interface advances, the solid-liquid interface will discharge solute to the liquid phase, which will affect the solidification process.Therefore,the effect of solute redistribution should be introduced in the simulation.

Based on Fick's Law[16], the solute field equation is introduced to realize the solute change in the computational domain during solidification:

whereCis mol fraction of salt in seawater,Dis dimensionless diffusion coefficient.

The phase field equation after adding solute is:

The temperature field equation is:

1.2 Solidification and melting model

In this paper, the solidification/melting model in Ansys-Fluent is used for numerical simulation of freezing stage of water droplet solidification[17].In this model, the enthalpy-porous medium technology is used to solve the solidification or melting problem at a certain temperature.By calculating the enthalpy balance of each cell in the computational domain,the liquid fraction of the fluid is obtained.When the liquid fraction is between 0 and 1, the solid and liquid coexist in the region.When the liquid fraction is 0, the fluid is completely solidified, and the porosity and velocity in the solid-liquid coexistence region are 0[18].

In the system,the total enthalpyhtotalof the material is:

wherehis apparent enthalpy,htotalis the latent heat of the material after solid-liquid mixing.

The latent heat after solid-liquid mixing is:

whereαliquidis the liquid fraction in the region where phase transition occurs, andLmixis latent heat for materials.

The solidification/melting model regards the phase transition zone as a porous medium,and its internal flow conforms to the Darcy flow law and Carman-Koseny assumption.Therefore,the source termS→Mof momentum equation for phase transition process is introduced:

whereεis the minimum value to prevent denominator from being 0,Amushis the coefficient of solidliquid mixing zone,which is related to the porosity of porous media.

2 Numerical method

2.1 Outline of water droplets

2.1.1 Water droplet profile in initial stage Fig.2 is the outline of water droplets at the initial stage.V0is the volume of water droplets,R0is the base radius,H0is water droplet height, andθ0is contact angle.

Before calculating the water droplet profile, the droplet is generally regarded in an ideal state, so several hypotheses need to be made:

(1) When the water drop is standing on the cold surface, it is rotating symmetrically,and its axis of rotation is vertical;

(2)The convective heat transfer between water droplets and air can be ignored;

(3)The volume of water droplets is not affected by evaporation or condensation.

Usually,the volume range of droplets in ship icing is 10-1000 μL[19].In order to meet the actual requirements and facilitate the calculation,the droplet volume of 10 μL is selected in this paper for subsequent water droplet solidification calculation.

For static water droplets,the base radius and water droplet height can be calculated by the volume and contact angle of water droplets.

Fig.2 Initial stage droplet profile

whereRNis the base radius,HNis the height of water droplets,Vis the volume of water droplets,andθis the wall contact angle.

In this research, the contact angles are 30°, 60°, 90°, 120° and 150°.Combined with the volume of water droplets, the initial contours of standing water droplets under different contact angles could be obtained.

2.1.2 Water droplet profile after recalescence stage

Fig.3 is the outline of water droplets after recalescence stage.At the end of the recalescence stage, a uniform mixture of ice and water is formed inside the droplets.Due to the different densities of ice and water, the volume water of the droplets is enlarged toVN, the height becomesHN,the radius of the substrate becomesR,and the contact angle remains unchanged[20].

It can be seen from Fig.3 that the outline of water droplets has changed after the recalescence stage, so the outline of water droplets need to be recalculated.The density of ice-water mixture after recalescence can be calculated by the proportion of ice phase and the density of ice and water.Then,the volumeVNof water droplets after recalescence can be calculated by the initial volume of water droplets and the density of ice-water mixture.

Fig.3 Contour of water droplets after the recalescence phase

whereρminis the mixing density of ice water,ρwis water density,ρiis ice density,andγiis ice ratio.

With the volume of the droplet obtained after the recalescence stage, it is substituted into Eqs.(1)-(2)to calculate the base radius and height of the water droplet after recalescence.

After the recalescence stage, the droplet has become a uniform ice-water mixture, and its physical parameters also change accordingly.To calculate accurately, its physical parameters need to be processed.The physical parameters of water droplets,such as latent heat,specific heat capacity and thermal conductivity,are recalculated by Eq.(13).

2.2 Solidification simulation condition

2.2.1 Parameter setting of recalescence stage

Setting square grid of 1000×1000 for phase field simulation, grid spacing Δxand Δyare dimensionless 0.03,time step Δt= 0.0001 s.

In order to ensure the convergence of the results, the time step and grid spacing should meet the following conditions[21]:

2.2.2 Parameter setting for cooling stage

The software used in the solidification model of water droplets is the solidification/melting model in Fluent.In the freezing stage,due to the limitation of the model,the contour change caused by the volume change in the process of water droplet freezing cannot be reflected.Therefore, the change of density with temperature is added to the physical parameters of the material,so as to calculate the volume change of water droplet freezing in the subsequent work.The density change in freezing is based on the density change of ice with temperature[22].In this paper, the test results are fitted, and the fitting formula of temperature and density is applied to the material attribute setting in Fluent.

whereρSis the ice density during freezing,andTSis the ice temperature.

The model used in this simulation is shown in Fig.4.The temperature of the cold surface is set as the freezing temperature, and the temperature inside the water droplets is set as the temperature of the ice-water mixture.Since the reference is the seawater with a salinity of 3.5%, the temperature inside the water droplets is set to 270.85 K according to the salt water solidification point.

Simulation calculation is transient calculation.Pressure-based solver is selected.The velocity and pressure are coupled by SIMPLE method.The momentum and energy equations are discretized by Second Order Upwind scheme with a time step of 0.1 μs.In order to save calculation resources and improve calculation accuracy, the mesh size of 0.02 μm is selected for subsequent simulation.Fig.5 shows the flow chart of numerical simulation.

Fig.4 Physical modeling and boundary demarcation

Fig.5 Numerical simulation flow chart

2.3 Model validation

In this paper, the solidification/melting model in ANSYS-Fluent software is used for simulation.In order to verify the accuracy and effectiveness of the model,simulation is carried out according to the experimental conditions ΔT= 16.5 K,θ= 90°, and the step size is set to 0.01 s.(where ΔTrepresents the supercooling andθrepresents the contact angle.)

As shown in Fig.6(a),Zhang[13]carried out the freezing process experiment of static water droplets on the cold surface.The experimental results are compared with those in the literature, as shown in Fig.6(b).It can be seen that the variation trend of the simulated liquid-solid interface is consistent with that obtained by experimental photography.

In order to quantitatively compare the simulation results with the experimental results in the literature, the desired interface height during the droplet freezing process is treated as infinite toughening (the desired interface height at any time/the desired interface height at the end of solidification and freezing).As shown in Fig.7, from the variation of the dimensionless phase interface height with time obtained by experiments and simulations in the literature, it can be seen that the two are generally consistent in trend.In terms of the final freezing time, the experimental value in the literature is 9.2 s,and the simulated value is 10.2 s,with a deviation of 9.8%.

From the shape and height of the phase interface,the simulation and experimental trends are in good agreement, indicating that the model can be used to simulate the freezing process of supercooled droplets.

3 Numerical results and discussion

Fig.7 Comparison of calculated and experimental values[13] of dimensionless plane interfaces

3.1 Ice phase ratio

3.1.1 Ice phase ratio under different supercooling

Fig.8 shows the proportional growth of ice phase at each supercooling.The growth rate of ice phase ratio is related to supercooling.The greater the supercooling,the greater the growth rate of ice phase ratio.There is a threshold for the proportion of ice phase in the recalescence process.When the proportion of ice phase increases to a certain value,it will no longer change.This is because,as the recalescence process continues to release heat,the temperature in the entire calculation domain continues to rise.After the final temperature reaches the freezing point, it cannot continue to rise,and ice crystals cannot continue to grow.The ice phase ratio reaching the threshold represents the end of the recalescence stage.At this time,a uniform ice-water mixture has been formed inside the droplet, and the temperature in the calculation domain reaches the freezing point at the same time.Tab.1 is the ice phase ratio of partial supercooling.

Fig.8 Proportion of ice phase changing with time

Tab.1 Proportion of ice phases for each degree of supercooling

3.1.2 Droplet profile under ice phase ratio

Tab.2 and Tab.3 show droplet height and radius under different supercooling and contact angles.From Tab.2, it can be seen that as the subcooling increases, the proportion of ice phase increases, and the height and radius of water droplets also increase.The height and radius of water droplets are consistent.It can be seen from Tab.3 that with the increase of supercooling degree,both the proportion of ice phase and the height of water droplets increase, but the radius of water droplets decreases.

Tab.2 Droplet height and radius at different degrees of supercooling

Tab.3 Droplet heights and radii under different contact angles

Fig.9 shows the droplet profile under different supercooling.Under different supercooling, the contour gap of the droplet is not obvious compared with the overall contour.This is because different supercooling only affects the ice phase ratio of the droplet,and the influence of the ice phase ratio on the volume of the droplet is not obvious compared with the original volume,so the supercooling has little effect on the contour of the droplet.

Fig.9 Droplet profiles at different degrees of supercooling

It can be seen from Fig.10 that when the contact angles are different, the contour of the droplet is quite different from the overall contour.This is because when the contact angle is different, the radius and height of water droplets have changed greatly, which directly affects the contour of water droplets.

3.2 Simulation results of icing

The supercooling degree is 30 K and the contact angle is 90°when the near order of water droplet freezing is simulated.The freezing completion time is 33.8 μs.The simulation results are shown in Fig.11.It can be seen from the figure that the solid-liquid interface is basically a straight line when advancing upward, only inside the droplet near the surface of the droplet.Due to the tension, the solid-liquid interface presents a certain bending angle.In the freezing stage, the initial solid-liquid boundary advances faster at first, then slows down.This is because the heat transfer efficiency is different before and after the whole process.In the initial stage of freezing, the solid-liquid interface is close to the cold surface.The heat transfer from the solid-liquid interface to the cold surface is faster, and the release rate of solidification latent heat is faster.After that, the solid-liquid interface is gradually moving away from the cold surface,and the distance becomes larger,and the release of latent heat slows down.At the same time,due to the different cooling rates in the droplet,the closer to the solid-liquid interface,the higher the temperature of the droplet, which leads to the temperature inside the droplet higher than that of the cold surface,and further inhibits the heat transfer efficiency.

Fig.10 Droplet profiles at different contact angles

Fig.11 Droplet freezing process at a subcooling degree of 30 K

3.3 Effect of supercooling on icing

Tab.4 and Fig.12 are the freezing time of droplets under different degrees of supercooling.It can be seen from Tab.4 that with the increase of supercooling degree, the freezing time is gradually shortened.The supercooling degree is changed from 45 K to 30 K, and the freezing time is shortened from 33.8 μs to 24.6 μs.It can be seen from Fig.9 that the larger the supercooling degree is,the smaller the reduction of freezing time is,which indicates that the supercooling degree has a significant impact on the freezing time in the freezing stage.After converting the supercooling degree into the temperature condition, it can be found that with the temperature increase, the relationship between freezing time and temperature is approximately exponential function.

Tab.4 Droplet freezing time at different degrees of supercooling

Fig.12 Droplet freezing time at different degrees of supercooling

3.4 Effect of contact angle on icing

The influence of contact angle on supercooled water droplets in freezing stage is simulated.A total of five contact angles of 30°, 60°, 90°, 120° and 150° are selected, and the supercooling degree is 30 K.

Fig.13 is the images of supercooled water droplets freezing process under different contact angles.It can be seen from the figure that under different contact angles, at the interface between the solid-liquid interface and the droplet surface,the angle between the interface and the outer surface of the droplet is approximately 90°.The solid-liquid interface inside the droplet is different in shape.When the contact angle is 30°-90°, the solid-liquid interface at the contact position of the interface and the outer surface is almost linear except the contact position.However, when the contact angle increases to 120°, although the solid-liquid interface is still advancing as a whole, the overall morphology of the solid-liquid interface changes from almost a straight line to a small bending.When the contact angle is 150°,the solid-liquid interface shows an obvious bending.

Fig.13 Freeze under different contact angles

Tab.5 and Fig.14 show the droplet freezing time under different contact angles.Droplets with different contact angles have great differences in freezing time.The contact angle increases from 30° to 150°, and the freezing time increases from 8.02 μs to 95.6 μs, increasing by 1092.02%.Compared with supercooling, the contact angle has a greater impact on the freezing stage of water droplets.The larger the contact angle,the longer the freezing time.This is because when the contact angle is small, the contact area between the droplet and the cold surface is larger, which is more conducive to heat conduction.At the same time, because the droplet height is small, the distance between the solid-liquid interface needing to be promoted is shorter, which further reduces the freezing time and makes the freezing faster.When the contact angle is large,not only the contact area between the droplet and the cold surface is smaller, which slows down the heat transfer efficiency,but also the distance between the solid-liquid interface needing to be promoted is longer,which leads to a longer freezing time.

Tab.5 Droplet freezing time at different contact angles

Fig.14 Droplet freezing time at different contact angles

With the influence of supercooling degree combined with contact angle on freezing time, it is found that supercooling degree and contact angle have a great influence on freezing time, and they are exponential function.

4 Conclusions

A numerical framework was established to consider the effect of the ice-phase ratio on the freezing of water droplets after the recalescence stage.By tracing the solid-liquid interface, the freezing process of static supercooled water droplets on a cold surface was studied.The influence of surface contact angle and subcooling degree on water droplet freezing was analyzed.The main conclusions are as follows:

(1) Under different subcooling degrees, the contour difference of water droplets is not obvious compared with the overall contour.When the contact angles changes, compared with the overall contour,the contour difference of water droplets is very obvious.

(2) The growth rate of ice phase ratio is related to supercooling, and the greater the supercooling,the greater the growth rate of ice phase ratio.

(3) The degree of supercooling and the contact angle have a great influence on the freezing of stationary droplets.Freezing time decreases exponentially with the increase of subcooling degree.When the supercooling is 30 K,the freezing time is 33.8 μs,and when the supercooling is 45 K,the freezing time is 24.6 μs.The influence of contact angle on the freezing time is also close to exponential function.The contact angle increases from 30° to 150°, and the freezing time increases from 8.02 μs to 95.6 μs.

This model can serve as a reference for the prediction of ice on the surface of superstructures.