Numerical simulation of a droplet icing on a cold surface

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1 Numerical simulation of a droplet icing on a cold surface Buccini Saverio Peccianti Francesco Supervisors: Prof. Alessandro Bottaro Prof. Dominique Legendre March 30, 2017 Università degli Studi di Genova

2 Outline 1. Introduction 2. Physical model and experiments 3. Numerical method 4. 1D problem and validation 5. Simulations of the droplet 6. Conclusions 1

3 Introduction

4 The icing problem - Eects Eects ˆ Lift variation stall ˆ Increased drag ˆ Instrumentation problems AF447 ˆ Increased weight 2

5 The icing problem - Countermeasures Countermeasures ˆ De-icing & anti-icing uids ˆ De-icing boots ˆ Electrical resistances ˆ Hot air bleeding from engines 3

6 Physical model and experiments

7 Freezing droplet Aim of the work ˆ Freezing front evolution ˆ Final shape of the droplet and its causes 1. Density variation 2. Marangoni eect 4

8 Stefan problem (1) ˆ solid phase: 0 x < s(t) T s t ˆ liquid phase: s(t) < x < = α s 2 T s x 2 s Boundary conditions T l t = α l 2 T l x 2 l ˆ Constant temperatures T (0, t) = T w and T (x, t) = T i ˆ Interface condition at x = s(t): T λ l T l x + λ s s x s + T s s = T s l + = T m s = ρ s L ds dt 5

9 Stefan problem (2) Assumptions ˆ Heat transfer is driven by the sole conduction ˆ Thermophysical properties constant with temperature in each phase ˆ Phase change temperature xed and known ˆ The entire domain initially at T (x, 0) = T i Analytical solution s = 2δ α s t T s (x, t) = T wall + (T ( m T wall ) x erf erf (δ) T l (x, t) = T i (T i T m ) erfc(δα) erfc 2 α s t ) ( x 2 α l t ) 6

10 Stefan problem (3) Adimensional parameters Ste = ρ sc p,s (T m T wall ) ρ s L φ = ρ lc p,l (T i T m ) ρ s c p,s (T m T wall ) T [ C] T 0 =-7 C T i =20 C t=10000s α = -5 αs -10 α l x Equation for δ e δ2 erf (δ) e δ 2 α 2 φ erfc(δα) α = δ π Ste 7

11 Velocity calculation Continuity ρ liq V liq + ρ sol V sol = const v liq = dh dt = ds ( dt 1 ρ ) sol ρ liq }{{} r Governing equation T l t + v T l liq y = α 2 T l l xl 2 Being the parameter r the indicator of the expansion: [ ( )] x erfc αδ 2δ α s t r T l = T 0 + (T m T 0 ) erfc [αδ (1 r)] e δ2 e (αδ)2 erf (δ) φ α erfc [αδ (1 r)] e 2r(αδ)2 e (rαδ)2 = δ π Ste 8

12 Numerical method

13 JADIM Jadim is a research code developed by J. Magnaudet and D. Legendre's team in the Interface group at Institut de Mécaniques des Fluides de Toulouse. The code permits to describe in an accurate way physical mechanisms present in multiphasic ows. ˆ Volume of Fluid formulation is employed ˆ Thermal and Immersed Boundary Method routine are supported In the present work the objective is to couple the three of them and, in particular, to develop a thermal based IBM formulation 9

14 Solid function fraction A solid function fraction has been dened as follows, representing the amount of ice for each cell: T max T α ibm,lin = τ T max T [ min α ibm,cos = 0.5 τ cos π (T T ] min) + 1 T max T min 1 ibm,linear 0.8 ibm,cos 0.6 ibm T [ C] φ = (1 τ) φ air + τ [(1 α ibm ) φ liq + α ibm φ sol ] 10

15 Implementation of velocity Velocity calculation v liq = ( 1 ρ ice ρ water ) Scalar transport equation: α ibm t V front + V front α ibm = 0 V front n = α ibm T n i = T t α ibm α ibm x i α ibm Velocity imposition f = α ibm U U t { U = 0 αibm 0.95 U = v liq 0 < α ibm < 0.95 Being U a predictor velocity without considering the immersed object Pressure correction Jadim's SIMPLE algorithm is modied in order to take account of the calculated velocities 11

16 The latent heat computation { H = cp,l T Liquid H = c p,s T + L Solid The source term method T ρc p t = ( k T ) S x x S = α IBM T T t [L + T (c p,s c p,l )] c p ρh t = (k T ) The apparent heat capacity method c app = dh dt c app = c p + L dα IBM dt ρc app T t = (k T ) 12

17 1D problem and validation

18 IBM functions err T,max err T,avg err int,max err int,avg Linear [ 2, 2] 6.1% 5.7% 27% 17% Linear [ 1, 1] 4.3% 4.1% 14.2% 9.2% Linear [0, 1] 5.2% 4.2% 4.7% 2.2% Linear [0, 0.1] 2.8% 2% 3.5% 1% Cosine [ 1, 1] 5.2% 4.3% 12.2% 7.6% Cosine [0, 1] 5.3% 4.1% 4.8% 2.3% ˆ For narrower solidication ranges, the results are more accurate ˆ The solidication front is well located if the inferior limit coincides to 0 C 13

19 IBM functions, gures Freezing front, cos [ 1, 1] Freezing front, lin [0, 0.1] 14

20 Velocity err vel,max err vel,avg Lin [ 1, 1] 16.5% 8.1% Lin [0, 1] 23.6% 8.7% Lin [0, 0.1] 100% 53.8% Cos [ 1, 1] 13.7% 7% Cos [0, 1] 23.6% 8.9% ˆ Velocity is generally underestimated ˆ The chosen range has to be wide enough to properly calculate the velocity 15

21 Grid convergence Figure 1: Temperature, cos [ 1, 1] Figure 3: Interface position, cos [ 1, 1] Figure 2: Velocity, cos [ 1, 1] Figure 4: Interface position, lin [0, 1] 16

22 Latent heat Figure 5: Apparent capacity method ˆ Average errors are comparable: 20% ˆ Source term method tends to become more accurate through time ˆ Apparent heat capacity worsens with time due to underestimation of the latent heat ˆ Source term method doesn't guarantee stability without additional loops Figure 6: Source term method 17

23 Simulations of the droplet

24 Parameters of the simulations V [ m 3] R 90 dx dt [m] 10 6 [m] 10 7 [s] [ α ] m 2 s [ ] ρ kg m 3 Air Water Ice ˆ West wall represents the cold plate and it is isothermal ˆ South is the symmetry axis ˆ North and east walls are adiabatic T i T w 18 C 5 C Bo = g ρ d 2 γ 1 18

25 The solidication process (1) 19

26 The solidication process (2) ˆ Higher thermal diusivity of the air causes the droplet outer layer to solidify ˆ Thermally treated as a mixture between ice and water ˆ Dynamically considered as liquid 20

27 Velocity eld ˆ Inside the solid velocity is zero ˆ Composition of V x and V y returns a vector perpendicular to the interface ˆ Velocity eld in the water comes from the combination of incompressibility and the axis of symmetry 21

28 Evolution of the freezing front (1) ˆ Quasi-linear trend till the outer layer of ice is thin ˆ Acceleration due to the increased thermal diusivity 22

29 Evolution of the freezing front (2) ˆ Experimentally the trend seems to be linear dierently from the Stefan problem ˆ An expanded domain guarantees a more accurate result, minimum size depends on the contact angle ˆ Numerical results conrm the global linear behaviour 23

30 Contact angle inuence (1) ˆ Wall temperature ˆ Initial temperature ˆ Droplet's volume ˆ Mesh size 24

31 Contact angle inuence (2) ˆ Hydrophobic surfaces tends to delay the ice accretion, commonly employed as constructive materials or coatings ˆ We observed no formation of the protrusion in correspondence of a threshold of about Θ 30 25

32 Contact angle inuence (3) ˆ Results similar to other numerical works ˆ Quasi-linear behaviour independent from the contact angle, the dierence is the total solidication time 26

33 Elliptical droplet (1) It is not a physical case the droplet's volume and consequentially Bond number are too small to cause an elliptical shape A qualitative study to investigate if new dynamics appear 27

34 Elliptical droplet (2) ˆ The freezing front shape resemble the spheric cap case and the experimental results ˆ The pointy protrusion continues to appear 28

35 Conclusions

36 Conclusions and future developments Conclusions ˆ Successfully coupled VoF, thermics and IBM ˆ The characteristic pointy tip has been reproduced ˆ Evolution of the freezing front conrms experimental results over analytical ones ˆ Parametric study of the contact angle inuence Future developments ˆ Ameliorate the computation of the latent heat of solidication ˆ Tracking the freezing front for a better calculation of the velocity ˆ Experimentally conrm the inuence of the contact angle 29

37 Thank you for your attention

Heat and Mass Transfer over Cooled Horizontal Tubes 333 x-component of the velocity: y2 u = g sin x y : (4) r 2 The y-component of the velocity eld is

Heat and Mass Transfer over Cooled Horizontal Tubes 333 x-component of the velocity: y2 u = g sin x y : (4) r 2 The y-component of the velocity eld is Scientia Iranica, Vol., No. 4, pp 332{338 c Sharif University of Technology, October 2004 Vapor Absorption into Liquid Films Flowing over a Column of Cooled Horizontal Tubes B. Farhanieh and F. Babadi