Deformation of freezing water droplets on a cold copper surface

Science in China Series E: Technological Sciences 2006 Vol.49 No.5 590- -600DOI: 10.001/51431-006-2017-yDeformation of freezing water dropletson a cold copper surface .WANG Jieteng, LIU Zhongliang, GOU Yujun, ZHANG Xinhua& CHENG ShuiyuanCollege of Environmental and Energy Engineering, Beiing University of Technology, Beiing 100022, ChinaCorrespondence should be addressed to Liu Zhongliang (email: liuzhl@bjut.edu.cn)Received March 22, 2006; accepted July 15, 2006Abstract Freezing processes of water and peanut oil droplets on a cold surface areinvestigated in this paper. We observed during our experiments that the base surface of awater droplet that is in direct contact with the cold surface keeps its original shape, but theother part of the droplet shows an obvious growth along the direction normal to the basesurface. One small protrusion appears on the top of the water droplet at the end of thefreezing process. The experimental observations also show that no obvious shape changehappens during the freezing of peanut oil droplets. It is postulated that the effects ofsurface tension and volume dilatation resulted from liquid-to-solid phase change cause theshape change and protrusions formation. Based on this postulation, a physical andmathematical model is developed. The results of the model of a water droplet s freezingprocess correspond with our experimental observations. The observed phenomenon thatfrost-growth speed on the protrusion is higher than that on the other part of the waterdroplet is also analyzed.Keywords: frost, droplet, freezing, protrusion formation, surface tension.Frost formation is a very common phenomenon in cryogenic, refrigeration, and airconditioning engineering. The temperature of heat exchange units will undergo a graduallydecreasing process during their thermal defrosting process. When humid air comes in contactwith a cold wall whose temperature is below the dew point of air, vapor contained in thehumid air will condensate and deposit on this cold surface in the form of small water droplets.Continuous condensation of vapor and glomeration of the small droplets will produce largerwater droplets. Meanwhile, when the temperature of the cold surface is lower than thefreezing point of water, the water droplets formed on the cold surface will freeze.Subsequently, frost will deposit on these frozen droplets. Frost deposition on heat transfersurfaces will produce a series of negative effects on heat transfer enuinment For example,heat transfer resistance and pressure loss will inc中国煤化工at transferperformance. In addition， the air passagMYHCNMHGndmal-www.scichina.com www.springerlink.comDeformation ofreezing water droplets on a cold copper surface591function of the equipment may occur.Hayashi et al.!" divided the frost formation process into three different periods, that is,the crystal growth, the frost layer growth, and the frost layer full growth period. Mostresearch works reported in literature are on the frost layer growth and the frost layer fullgrowth period, but some are concerned with the influences of the external electricalfield2l and the surface characteristics'- 5 on frost formation. Few work has beencompleted on the deformation of freezing water droplets on cold surfaces. Since theprocess is very fast, it has been neglected. However, this period, i.e. the frost crystalgrowth period, is not only the starting point of the frost deposition process and should befully understood as the initial condition for numerical simulation, but it also relates tovery important theoretical problems of nucleation and phase change theory. Therefore,this research has significant relevance to frost formation research. .Feuillebois et al.tol studied the freezing heat transfer problem of supercooled waterdroplets of spherical shape. Richard and Mochel"T gave a detailed thermodynamicsanalysis of the same problem. Tabakova et al.s investigated the freezing process 0supercooled droplets with an assumption that the droplet surface temperature was equal tothe surrounding air temperature. All these research works are about the spherical droplet,which is not the same as the truncated spherical droplets formed on a cold surface. As faras the present authors know, the volume expansion has not been taken into considerationin any of the research reported so far. The shape of the water droplets formed during thefrost formation process is that of a truncated sphere that changes during the freezingprocess. Actually, the shape change of the water droplet during freezing is significant asthe protrusion may appear due to the density discontinuity of liquid water and solid water.Hindmarsh et al.! verified that a single droplet adhered to a thermocouples and put incold air changed its original spherical shape and formed a protrusion on the top side ofitself. The researchers attributed the protrusion to volume dilatation. They attributed thechanging from clear to opaque of the droplet to air bubbles released during freezingprocess. Wang et al. ' '“”analyzed the condensation of vapor and aggregation of droplets oncold surface. The number of droplets increased with the humidity of the air. A hexangularstructure was observed when the diameter of droplet exceeded critical size (80 μm). Butdeformation and protrusion formation were also not taken into consideration. Satoh etal." did not find protrusion formation in their droplets freezing experiments either. Ismailand Salinas2l gave a detailed model of frost formation, but no protrusion was mentioned.Kawanami et al.!15l observed the droplets' freezing process on cold surface and simulatedthe process by using Landau' s theory. Their simulation results did not agree with theexperimental observation of the top part of the droplet. They also did not take protrusionformation into consideration. Shape change and protrusion formation at the top of dropletsat the final stage of freezing was observed by Hokel4. He used a high speed camera toobserve the frost deposition process. He explained that the shape change and theprotrusion were due to the interior convection of the droplatH the niffronce between中国煤化工。specific volumes of ice and water. He also noted thed only onhydrophilic surfaces and not on hydrophobic surfaceHCNM HGIt was found in our former researchestsl that there were different frost growth speeds592Science in China Series E: Technological Sciencesat the different positions after the droplet froze under free convection condition. Thespeed on the top was the fastest. Under preferable conditions, an ice column formed onthe top of the droplet. This occurrence is often called ice column growth and is shown inFig. 1. The column growth was obvious after 15 min of growth. It was observed that atthe end of the freezing period that a protrusion formed on the top of the droplet that waslying on a hydrophilic copper surface. The ice column growth always appeared at thislocation.Fig. 1. Ice column formed after droplet freezing. ta= 19.3 ,0= 69%,1w=-9.31 Experimental apparatus and procedures1.1 Experimental apparatusThe objective of this research was to investigate the freezing process and deformationof droplets experimentally and numerically, and to seek the mechanism of protrusionformation. The experimental apparatus that was used is shown in Fig. 2. The test surfacewas a copper cold plate (4) with a dimension of 150 mmx52 mmx6 mm. The cold platewas connected to the thermoelectric cooler (3). The temperature of copper plate surfacewas measured using 4 Type-T thermocouples (5), which were buried in 4 holes drilledinto the plate. These holes were 0.5 mm in diameter and 13 mm in depth. The locationsof the thermocouples are shown in Fig. 2-I. The copper plate and electric cooler could beset vertically or horizontally to research the effects of convection on droplet deformation.Lowest working temperature of the thermoelectric cooler made by Tianjin ZhonghuanElectrical Refrigeration Research Institute was -26C. Water in the feed pipe and in theexhaust pipe was used as cooling medium to keep the equipments (2) and (3) workingnormally. Microscope (7) with enlargement factor of 7-112.5 and camera (8) and CCD(9), manufactured by Beijing Glasses Instrument Factory, were used for observation andmeasurement of the droplet size and its freezing process. The image obtained wasrecorded by a computer (10). The software to acquire the images was called “BoserBS602”. The processing software used to get dimensions of the droplets was MrV nt300which had been calibrated by a micrometer before testing. The temperature of the air andthe cold surface were recorded by means of a data acquisition system (Agilent 34907A)that was connected with the computer. The conta中国煤化工a contactangle analyzer (FTA125). A cold light iluminator wIn order to isolate water droplet from humid air,:TYHCNMHG laid overDeformation ofreezing water droplets on a cold copper surface593the cold plate and nitrogen of 99.999% purity was constantly charged into the enclosure,as shown in Fig. 2-II. The purpose was to prevent the water vapor condensing on thedroplet and forming frost crystals, thus identifying the possible influences of humid airon water droplet freezing.13,↓26;↓39-1I8「丁7I 999%9N,12Plexiglass enclosure_ IIntake port2Water dropletCold plateFig. 2. Scheme of the experimental apparatus. 1, Water source; 2, electric source; 3, thermoelectric cooler; 4, coldcopper plate; 5, thermocouples; 6, data acquisition board; 7, microscope; 8, camera; 9, CCD; 10, computer; 11, cable;12, water inlet; 13, water outlet.1.2 Experimental procedureA water droplet of about 1.0 mm in diameter was put on the cold copper plate surfaceby a special injector. The temperature was set at the given value on the thermoelctriccooler. The microscope and image recording systems were adjusted to work properly forimage recording and data acquisition. Tests were carried out under the followingconditions with the air temperature of 17.0^C:1) Variable surface temperature freezing. The cold plate surface temperature was firstset to 0C and after its steady state achieved, a drop of water was put onto the cold platesurface. At the same time, the set value of the surface temperature was changed to -10Cand thus a gradual cooling process was started. The freezing process of the water dropleton the cold copper plate was actually accomplished with a variable cold surfacetemperature. The cold plate surface temperature recorded was thus time dependent andwas related to time by the following equation:tw =-1031 + 10.09exp( -0.00794t)，τ≥0.(1)Here, τ is time (s), tw is the cold copper surface temperature (C).2) Constant surface temperature freezing. The cold plate surface temperature was setto -10C and after its steady state achieved, a drop of water was put onto the cold platesurface. Therefore, the freezing process of the water droplet on the cold copper plate wasactually accomplished with a constant cold surface t中国煤化工3) The freezing process of peanut oil and water d| YHCNMHGhesameasthe variable surface temperature freezing, but in order to study the posslble Influences of594Science in China Series E: Technological Sciencesair moisture on the freezing process, nitrogen was used to isolate the droplet from humidair.All of these experiments were carried out under the air temperature of t:= 17.0C.2 Experimental phenomena and analysis2.1 Observation of experiments(i) Variable surface temperature freezing process. The initial shape of the dropletlying on the cold copper surface at temperature of 0C is of truncated sphere due to theeffect of surface tension. The diameter of the droplet and its dynamic contact angle withthe copper surface are 0.75 mm and 72° , respectively. Fig. 3 presents a group of picturestaken of the droplet during freezing. The droplet is transparent before freezing (Fig. 3(a)).However, when the cold plate surface temperature was lowered to about - -7.0C, thedroplet lost its transparency suddenly and turned to fully opaque (Fig. 3(b)). The timelapse between the last clear water state of the droplet (Fig. 3(a)) and its fully opaque state(Fig. 3(b)) was very short, estimated to be less than 0.08 s, which is the interval of twoadjacent frames of the camera. The part of the droplet in contact with the cold surfacefirst cooled down then changed into ice. A mushy freezing front, indicated by an arrow inFig. 3(b), was observed. It moved continuously toward the top until the whole dropletwas frozen and a protrusion appeared on the top (Fig. 3()). The interval (between (a) and(C)) was 0.96 s. After that, frost appeared at the protrusion first, and then it becamevisible in other locations after 15 min of testing (Fig. 3(d)).0.5 mm| (c)(d)↓Fig. 3. Freezing process of droplets under variable surface temperature. t=17.0'C, δ= 10%, tw as eq. (1).(i) Constant surface temperature freezing process. A water droplet with initialtemperature of 17.0°C is directly put onto the中国煤化工ice whosetemperature is -10C. The droplet is more like asMHC N M H Gitact anglewith the cold surface than that of variable surface temperature freezing. It should beDeformation ofreezing water droplets on a cold copper surface595noted, from Fig. 4(a), that the freezing front is clearer than that of variable surfacetemperature freezing. In addition, the part above the distinct freezing front is transparent,and the part below the front is opaque. The freezing front moves to the top with time. Theshape of droplets after freezing is show in Fig. 4(b). Total time needed for this processwas 2.56 s, which is obviously longer than the variable surface temperature experimentbecause cooling of droplets is needed before its freezing， and the protrusion is moreobvious. It is also observed that the frost growth speed on the protrusion is much fasterthan that of the other part of the frozen droplet in subsequent process.(b)Fig. 4. Freezing process of droplets under constant surface temperature. t=17.0'C, 0=10%, Iw=-10'C.(ii) The freezing process of peanut oil and water droplets. In this group ofexperiments, the same procedure as for the variable surface temperature freezing wasrepeated for peanut oil droplets, instead of water droplets. Fig. 5 shows the comparison ofa droplet before (Fig. 5(a)) and after (Fig. 5(b)) freezing. From Fig. 5 we can see thatthere were no protrusions formed. The only phenomenon that indicates solidification wasthe transparency change of the droplet. After the completion of the freezing process, thedroplet was changed from clear to fully opaque.b)Fig. 5. Comparison of peanut oil droplet before and after freezing. I=17.0C, δ= 10%, tw as eq. (1).The experiments were also conducted on the freezing process of water droplets in theenvironment of pure nitrogen to detect the possible influences of water vapor in the air.The results of the experiments showed that there were no visible effects of airborn watervapor on the freezing process of water droplets when the water droplets were isolatedfrom the air by nitrogen. Further more, it did not matter whether the cold plate was setvertically or horizontally; the protrusion always formed on the top of the droplet (thefurthest end of the droplet from the cold plate).2.2 Analysis中国煤化工In our first group of experiments, that is, theMHCNMHGlesurfacetemperature freezing process, no phase changes were observed when the cold plate596Science in China Series E: Technological Sciencessurface temperature was equal to or slightly below 0C. The droplet suddenly becameopaque as the plate surface temperature was lowered to -5- - 10"C, which was typical ofthe supercooled state of water. The volume expansion caused by the phase change fromwater to ice near the cold plate surface could induce a large number of crystal embryosimmediately, thus make the droplet lower its transparency very quickly. The freezingfront was an approximately flat surface. A parameter, deformation factor (S), was definedto measure the deformation extent.s= H/H,Here, H is the height of a droplet at any instant, and H is the initial height before freezing.From Fig. 6, one can determine that the shape changes slow down during the onset stageto the completion stage of the freezing. However, during the period between the onsetand the completion of the freezing, the phase change rate keeps constant. No shapechange exists after the freezing finished.1.16 ._1.14马1.12-国1.10-s 1.081.06员1.04-1.021 .00+-4”140.0 140.2 140.4 140.6 140.8 141.0 141.2 141.4Time (tis)Fig. 6. Variation of deformation factor with time.For the second group of the experiments, that is the experiments for constant surfacetemperature freezing, the region with a temperature that was higher than 0°C in thedroplet would not freeze. So the phenomenon of ice embryos appearing in a whole watersphere instantly will not happen. A clear boundary, one side is opaque and the other sideis transparent, was found in the droplet. The total process included freezing anddecreasing of the droplet' s temperature. The time needed for the process is clearly longerthan variable surface temperature freezing.It was understood that in most matters, the phase change from liquid to solid usuallyhappened with compression or without volume change. Water is the only exception. Itexpands during the liquid-solid phase transition process. The above experimentalobservations prove that volume expansion caused by freezing is the only reason for thedroplet deformation and the protrusion formation during solidification. Internalconvection inside the droplet, the condensation of water vapor from air, and the frostdeposition have neglected effects.中国煤化工3 Mathematical modelTo simplify the analysis, several assumptions wel.MHCN MH G。ii uioity Uf the water .Deformation offreezing water droplets on a cold copper surface597droplet is variable during freezing although the total mass of liquid water and ice keepconstant. b) Water droplets on the cold plate surface are of the shape of a truncatedperfect sphere before freezing if gravitational effects are not taken into consideration. c)The frozen part keeps the final shape unchanged during the later process. The non-frozenpart is the shape of a truncated perfect sphere with a different origin from the initialdroplet due to the effect of surface tension. d) The immigration speed of the ice-waterinterface in the normal direction of the cold plate surface is a constant over the wholesurface of the droplet. The coordinate system used is depicted in Fig. 7. The diameter ofthe initial base surface of the droplet is 2ao; z is normal to the cold plate surface and theaxis of the droplet.Freezing frontE 9_MO.0 mN(0,0, m)00, 0, 0)Fig. 7. The coordinate system.We can suppose that the freezing front, a plane through the intersect circle of water-iceinterface and the outer surface of the droplet, is located at z =m at a time, that is,Z= m.(2)Then, the cross point of the freezing front defined by eq. (2) on the z-axe is M (0, 0, m).The freezing front plane is moving towards the top of the droplet until the droplet is fullyfrozen. The ice-water interface with temperature of 0°C is neither flat nor parallel withthe cold surface. The immigration speed of the phase change front is smaller in thecentral part of the droplet than in the other part. Actually, the part near the externalboundary of the water droplet freezes first. Therefore, the droplet can only expand towardthe top. As the cooling process proceeds, the ice-water interface moves toward the top.The shape of the part with zm changes its form while keeping a part of a standard sphere due to the effect ofsurface tension. The origin of this standard sphere is supposed at N(0, 0, n) with a .radius of r. Of course, n and r can be obtained from the mass conservation relationship.The heat transfer equation isItρcdt=λV2r+qy.(3))tHere, density ρ，specific heat C, and thermal conductivitv λ of water and ice are alldifferent. The droplet temperature varies with both中国煤化工e the frontsurface is a function of time. Source term qv is a res|YHC N M H Glatent heatrelease at the phase change interface. The location of the freezing front, m, can be gotten598Science in China Series E: Technological Sciencesby solving eq. (3) and by judging where the cell whose temperature equals to the phasechange temperature of water is. The heat transfer between the droplet and thesurrounding air is given by the following equation:一=a(t-1a)s.(4)Here, uu is the normal exterior of the droplet' s surface, s is the droplet s surface, and ta isthe air temperature. The temperature of the cold plate surface is given by eq. (1).If we suppose that the mass below the freezing front is Mi and the mass above thefreezing front is M2, thenM2= M -M;(5)=M -πj”" px2dy,M is the total mass of the droplet.The volume for a truncated sphere is calculated from the following equation:M2=ρ∠h(3a2 +h2),(6)where h is the height of the truncated sphere and a is the radius of its base circle. Fromeq. (6), one can easily solve for h, .6M,6M2(7)200wherep=x20ρ. a’The unfrozen part above the freezing front of the droplet is a shape of a truncatedsphere. The origin of the sphere is located at N(0, 0, n) and its radius is r. Therefore, thesurface equation of the sphere isx2+y2+(z-n)2=r2,(8)where n and r are given by the following equations:h2+a2(9)n=m+h-r.(10)The property used in the simulation is listed in Table 1.，Table 1 Property of ice and water used in the simulationMaterialDensity (kg/m')Thermal conductivity (kJ/C)Specific heat (kJ/kg. CWater998.20.64.182Ice915-0.01812+0.00781T)a) T is the ice temperature with the Kelvin degree. The melting heat of ice is (333 kJ/kg).中国煤化工4 Results and discussionMHCNM HGThe freezing process of a droplet with a base surtace diameter of 0.75 mm and aDeformation ofreezing water droplets on a cold copper surface599dynamic contact angle of 72° is numerically simulated, using the mathematical modeldeveloped previously. The numerical results show that, as one may expect, the shape ofthe droplet changes continuously until the droplet is fully frozen. Fig. 8 compares theshapes of the droplet before and after its freezing. As can be seen in this figure, the finalshape (solid line) of the droplet is quite different from its initial shape (dash line), thefrozen droplet is bigger than its initial liquid form. Fig. 9 is given to present the detailedshape at the top of the frozen droplet. As one can see, a protrusion does exist at the top ofthe droplet, which is in perfect agreement with our experimental observation (points).0.33 7After feezig0.30-0.25.0.20-Before feeing0.15-0.31 .0.10-至0.050.00003000.050.100.150.200250.300.350.40-0.100.000.050.10Radius a (mm)Fig. 8. Shape change before and after freezing.Fig. 9. Protrusion at the top of the frozen droplet.According to heterogeneous nucleation theory!16, the critical radius and critical Gibbsenergy change are given as2σ(11)△G。=.1682’coB )(θ cos ) θ(12)3g,where 8v is the Gibbs energy change between the parent phase and the new phase duringphase transition, and σ is interfacial energy. From these equations, one can find that thesmaller the contact angle (θ), the less the critical Gibbs energy change (OG,). The criticalGibbs energy change and the critical radius at protrusion are the smallest, which explainswhy the frost growth at the protrusion has the largest speed.5 Conclusions .The droplets do not keep their initial liquid droplet shape and grow in the directionnormal to the cold solid surface when they undergo a freezing process. Our experimentaland calculation results show that the deformation factor of water droplets caused byfreezing is 1.15- -1.20. The experimental observation also shows that the freezing of thedroplet will not take place until the water droplet is cooled to a highly supercooled state.At such a state, a slight disturbance will result in a large quantity of crystal nuclei, whichcauses a very fast phase change. The freezing front中国煤化工face of thedroplet that is in direct contact with the cold surHC N M H G protrusionappears at the end of this process, which makes the frozen droplet have a peach shape.600Science in China Series E: Technological SciencesThe specific volume difference between ice and water is the only reason for theprotrusion. Frost crystals grow faster at the protrusion than anywhere else, due to thesmallest nucleation barrier.Acknowledgements This work was supported by the National Natural Science Foundation ofChina (Grant No. 50376001)， the Key Project of National Fundamental Research andDevelopment (Grant No. G2005CB724201).References1 Hayashi Y, Aoki A, Adachi S, et al. Study of frost properties correlating with frost formation types. ASME JHeat Transfer, 1997, 99: 239- -2452 Wang C C, Huang R, Sheu E J, et al. Some observations of the frost formation in free convection: with andwithout the presence of ectric field. Intemational Jourmal of Heat and Mass Transfer, 2004, 47: 3491- -35053 Shin J, Tikhonov A V, Kim C. Experimental study on frost structure on surfaces with different hydrophilicity:density and thermal conductivity. ASME J Heat Transfer, 2003, 125:84- -944 Lee H, Shin J, Ha S, et al. Frost formation on a plate with different surface hydrophilicity. International Journalof Heat and Mass Transfer, 2004, 47: 4881- -4893Min J C, Webb R L, Benmisderfer C J. Long term performance of dehumidifying heat exchangers with andwithout hydrophilic coatings. Int J Heating, Ventilating, Air Conditioning & Refrigeration Research, 2000, 6(3): ;257- -2726 Feuillebois F, Lasek A, Creismeas P, et al. Freezing of a subcooled liquid droplet. Journal of Colloid andInterface Science, 1995, 169: 90- -1027 Richard R V, Mochel J M. Thermodynamics of melting and freezing in small particles. Surface Sciences, 1995,341:40- -508 Tabakova S, Feuillbois F. On the soldification of a supercooled liquid droplet lying on a surface. Jourmal ofColloid and Interface Science, 2004, 272: 225- -2349 Hindmarsh J P, Rusell A B, Chen X D, et al. Experimental and numerical analysis of the temperaturetransition of a suspended freezing water droplet. International Journal of Heat and Mass Transfer, 2003, 46:1199-12130 Wang C C, Huang R T, Sheu W J, et al. Some observation of frost formation in free convection with andwithout electric field. International Journal of Heat and Mass Transfer, 2004, 47: 3491- -350511 Satoh I, Kazuyoshi F, Hashimoto Y. Freezing of a water droplet due to evaporation-heat transfer dominatingthe evaporation- freezing phenomena and the effect of boiling on freezing characteristics. InternationalJournal of Refrigeration, 2002, 25: 226- 23412 Ismail K A R, Salinas C s. Modeling of frost formation over parallel cold plates. International Journal ofRefrigeration, 1999, 22: 425- -44113 Kawanami T, Yamada M, Fukusako S, et al. Solidification characteristics of a droplet on a horizontal cooledwall. Heat transfer- Japanese Research, 1997, 26(7): 469- -483.14 Hoke J L The Interaction between the Substrate and Frost Layer through Condensate Distribution. DoctoralDiseraion. University of llinois at Urbana Champaign. The Graduate College, 2000. 1015 Zhang X H, Liu Z L, Wang J T, et al. Experimental investigations on the influences of electric fields on frostlayer growth under natural convection conditions. Progress in Natural Science, 2006, 16(4):410- -41516 Byeongchul N, Ralph L W. A fundamental understanding of factors afcting frost nucleation. IntermationalJounal of Heat and Mass Transfer, 2003, 46: 3797- -3808中国煤化工CNMHG