14 8 Freezing droplets

Transcription

1 8 Freezing droplets

2 Task Place a water droplet on a plate cooled down to around -20 C. As it freezes, the shape of the droplet may become cone-like with a sharp top. Investigate this effect. 2

3 Equipment 3

4 Droplet Volume Contact angle Surface Inclination Curvature Material θ Temperature 4

5 Dependence on the volume of the droplet 14 Essentially the same 5

6 Inclination of the surface 6

7 Inclination of the surface 7

8 Curvature of the surface 8

9 Curvature of the surface 9

10 Curvature of the surface 10

11 Material of the surface 11

12 Contact angle of the droplet θ Different heights + Droplets of the same volume 12

13 Different contact angles 13

14 Temperature 14

15 Extreme volume of the droplet 4 cm 15

16 Summary of preliminary experiments Changeable parameters Droplet NO Volume QUALITATIVE Surface Inclination Curvature Material Contact angle Temperature CHANGES 16

17 What is the shape of the peak? Described by r r r 17

18 Existing literature ice-drops: How water freezes into a singular shape J.H. Snoeijer and P. Brunet, Am. J. Phys. 80, 764 (2012) Heat conduction equations solved Assumption: Planar freezing 18

19 Existence of reservoir Water blown away during freezing 19

20 Existence of reservoir Cut in half 20

21 Existence of reservoir Depression in ice 21

22 The peak: An alternative approach Shape as we approach the top: always a cone r 1 st r 0 no peak 0 peak occurs 22

23 Stabilized freezing: 14 water ice Vls V R Freezes to Vss V R V V V V ls R water ss R ice Vls volume of seen liquid But V we volume need of seen to solid know ss Equation for the VR volume shape of reservoir of reservoir 23

24 Heat flow on the water-ice interface Heat flow: perpendicular to the water-ice interface near the surface: parallel to the surface Interface is perpendicular to the surface 24

25 Shape of the reservoir Heat flowing unevenly: Hotter/colder areas are created Heat flows to the colder Heat flow becomes evenly distributed Stabilized freezing: Heat flow evenly distributed; perpendicular to interface Reservoir: spherical cap 25

26 Stabilized freezing: 14 water ice Vls V R Freezes to Vss V R V V V V ls R water ss R ice Substitute for V, V, V ls ss R 26

27 Volume of liquid and solid V ss r 3 3 tan V ls r cos sin 2 2 cos sin 27

28 Equation for alpha ice water 1 cos sin 2 2 cos 1 sin sin cos 1 sin tan cos sin cos 2 sin cos Solution: 25 28

29 Measured angle of approximately 25 on our droplet 25 29

30 Seemingly different droplets

31

32 Simulation Heat conduction Rotational symmetry No heat transfer to the air Plate: constant temperature 32

33 Simulation time increment Droplet is divided into segments with thickness h, contact area S Heat conduction between segments, thermal conductivity k Layer freezing L Q l ice ks of new ice added; latent heat of T h t l ice Q LS ice 33

34 Simulation 34

35 Real droplet vs simulated one 35

36 Freezing: Simulation vs. Experiment 36

37 Freezing: Simulation vs. Experiment 37

38 Summary of preliminary experiments Changeable parameters Droplet NO Volume QUALITATIVE Surface Inclination Curvature Material Contact angle Temperature CHANGES 38

39 Conclusion 39

40 Thank you for your attention Yes the snowman is made from frozen droplets r=cca 3mm 40

41 Apendix 41

42 Hairiness 42

43 Hairy droplets known problem 43

44 Volume of solid v V s r 3 2 v v tan r v r tan r V s r 3 3 tan 44

45 Volume of liquid R h r 2 h V 3R h l 3 r tan R h r h R tan r sin R r R sin V l r cos sin 2 2 cos sin 45

46 Spherical shape of the droplet ρ : liquid density g : gravity acceleration ρgr 2 R : diameter of perfect sphere with the same volume γ surface tension γ 46

47 47 14

48 48 14 Solidifying sessile water droplets W. W. Schultz, M. G. Worster, D. M. Anderson

49 49 14 The shape of solidifying droplet when ρ = 0.9

50 50 14 Close up

51 V vl r cos sin 2 2 cos sin V vs 3 3 r tan r 90 r P V V vl vs V V R R V R r sin cos 2 2 sin cos 51

52 α ,2 0,4 P 0,6 0,8 P

53 Shape of reservoir r r 53

54 Planar freezing V PV l S V l r 3 V V l s volume volume of of liqiud solid P s Using known formulas for volumes Spherical cap 3 1 cos sin 2 2 cos sin Cone 3 V s r tan 3 54 l

55 Planar freezing V PV l S V V l s volume volume solid We can calculate density/pike angle relation of of liqiud P s l P 2 1 cos 2 cos sin sin tan 55

56 α 14 Angle of cone vs ratio of densities Existence of cone only for P < 0, cos 2 cos sin sin P tan ,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 P 56

57 Simulated droplets with different P 57

58 Simulation vs theory α Simulation Our theory 0 0,5 1 1,5 P 58

59 Simulated extremes - size 3 times smaller 10 times larger Change of peek angle is negligible 59

60 Simulated extremes - temperature -12 C -60 C Change of peek angle is negligible 60

61 Simulated extremes 34kJ heat capacity 14 Change of peek angle is negligible 61

62 Simulated extremes contact area radius of 14 r=2,5 mm r=4 mm Change of peek angle is negligible 62

63 Simulated extremes contact area r=5 mm radius of 14 Change of peek angle is negligible 63

64 Simulated extremes conductivity 4 times larger 14 Change of peek angle is negligible 64

65 Simulated extremes all together Change of peek angle is negligible 65

66 Density ratio solid liquid 1 it works Bismuth (Bi) Antimony (Sb) Silicon (Si) Germanium (Ge) High temperature of melting Gallium (Ga) Toxic Water 66

67 STARE SLIDY 67

68 Existing literature ice-drops: How water freezes into a singular shape J.H. Snoeijer and P. Brunet Am. J. Phys. 80, 764 (2012) Planar freezing Ratio of densities Critical ratio of densities to create convex pike is ¾ Water is not predicted to create convex pike 68

69 Formation of pike for different 14 density ratios linear freezing Water P = 0.65 OBVIOUSLY WRONG P = 0.75 liquid P = 0.85 P=0.9 P = 1 P = 1.2 P solid Result => P 0.75 => pike 69

70 Middle section of a freezing droplet Assumed shape Real shape Exact shape of reservoir? 70

71 Middle section of freezing droplet Assumed volume Real volume V V V PV V l PV S ls R ss R V ls volume of seen liquid V ss volume of seen solid V R volume of reservoir 71

72 α 14 Angle/density ratio FOR P<1 => PIKE IS CREATED WATER 0 0,2 0,4 0,6 0,8 1 1,2 P 25 72

73 Simulation of the process of freezing Heat convection Changeable parameters Density ratio Volume of droplet Contact area Temperature of plate Heat capacity Heat conductivity 73