Multiphase Flow and Heat Transfer

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1 Multiphase Flow and Heat Tansfe ME546 -Sudhee Siddapueddy

2 Pendant and Sessile Dops Pendant doplet detachment sequence Sessile wate doplet immesed in oil and esting on a bass suface

3 Inteface Shape at Equilibium Conside a sessile dop sitting on a smooth solid suface inside anothe fluid. Fluid II Fluid I θ

4 Inteface Shape at Equilibium In the absence of gavity, the dop takes spheical shape with least suface aea. The dop is defomed by gavity. The cente of mass of the dop is foced to be loweed by gavity. This inceases the suface aea, which is opposed by suface tension foce. Assume thee ae no extenal foces acting on the dop. Fluid II Fluid I θ

5 Inteface Shape at Equilibium Oigin of the coodinate system is O, located at the apex of the suface. The dop is assumed to be axisymmetic. At O: Radii of cuvatue: = = At any P: Radii of cuvatue ae and Fluid II Fluid I O α P X θ α Q Z

6 At oigin using Young-Laplace: P I P II At point P: P I PII p Fluid II θ Fluid I O α α P X Hydostatic pessue heads at any point (P) on inteface as seen fom fluids I and II: Q Z P P I p IIp PI g I PII II g x sin

7 Bashfoth-Adams Equation g x I II sin Bo g II I Bo sin x sin g x II I Bashfoth-Adams Equation

8 Bashfoth-Adams Equation sin Bo x Bo = f (d/dx, d /dx ) Numeically solvable with appopiate BCs. I II g Solution in tabula fom (Bashfoth and Adams in 883) - Vaiation of / and x/ with α at a given Bo

9 Bashfoth-Adams Equation xmax max Pofile of dop pedicted by Bashfoth-Adams equation at Bo = 5

10 Bond Numbe Bo I II g Ratio of the gavity foce to the foce due to suface tension. Bo << Bo >> Dop will not defom significantly Lage defomation of the dop Bo <<, if the dop is small o the intefacial tension is lage o the density diffeence between the two liquid is low

11 ρ I > ρ II ρ I < ρ II Bo Bo >, dop shape is oblate I Weight of the dop flattens the suface Eg: Rain dop, dop on a suface Bo <, dop shape is polate Buoyancy elongates shape vetically Eg: Vapo bubble in liquid II g

12 Shape of Raindops Bo I II g L c g l v Bo L c << L c Gavity is negligible L c? mm, ai-wate inteface at 5 C

13 Shape of Raindops Bo L c.7 mm, ai-wate inteface at 5 C mm, Nealy pefect sphee >> mm, inceasingly flattened L c > 4.5 mm, Raindop beaks into smalle dops due to inteaction with ai. Falling aindops ae defomed by the inteaction with the ai and neve take the familia teadop shape with a pointed tail and a ounded bottom head.

Small aindops ae dominated by suface tension (nealy spheical) Lage aindops

14 Shape of Raindops Steadily falling aindops ae subject to the combined effects of suface tension, gavity, fiction, and ai cuents. Small aindops ae dominated by suface tension (nealy spheical) Lage aindops assume a typical hambuge shape Hee is slightly geate than L c (.7 mm) and the dop is slightly oval

15 Liquid Climbing the wall Z Shape of a fee liquid suface meeting a plane vetical wall. Solid Gas (Fluid II) If the liquid wets the wall (θ < 9 ), the liquid level will ise as the wall is appoached, meeting the wall at θ. θ Liquid (Fluid I) X Conside a D configuation:

16 Liquid Climbing the wall P I P P P I II II P P g v g l Solid = climb = Z θ (x) Gas (Fluid I) Liquid (Fluid II) X l v g d dx x cot

17 Radius of Cuvatue x a b R c Diffeentiating w..to x d dx x a b Howeve tan d dx tan x a b

18 Radius of Cuvatue cos tan b R c d cos d R c d cos d d d d dx dx d d dx d dx R c d d dx dx 3

19 3 d dx d dx l v g l v Multiplying by ' g 3 l v g 3

20 Integating BCs:, ' = as x Integal constant, C = C g v l 3 dx d Since,

21 g v l The height to which the liquid climbs at the vetical wall cot x climb sin g v l x Liquid Climbing the wall

22 The shape of the inteface: Liquid Climbing the wall c climb c climb L 4 L 4 cosh cosh L L L x c c c g L v l c

23 Liquid Climbing the wall

24 Capillay Rise o Depession A cylindical containe is filled with satuated liquid R-34a and its vapo at 3 C. Detemine the height to which the liquid will climb the vetical walls of the containe if the contact angle with the walls is 5. At 3 C, σ =.7 N/m, ρ v = 39.8 kg/m 3, ρ l = 8 kg/m 3. climb sin l v g climb.mm

25 Maangoni Foces Any vaiation in suface tension along an inteface will ceate tangential (shea) foces, known as Maangoni Foces. This vaiation can aise fom inhomogeneous mateial popeties, o fom tempeatue vaiations. Unless balanced by othe foces, these shea foces cannot be sustained in a liquid at est - will set it into motion.

Liquid gas interface under hydrostatic pressure

Liquid gas interface under hydrostatic pressure Advances in Fluid Mechanics IX 5 Liquid gas inteface unde hydostatic pessue A. Gajewski Bialystok Univesity of Technology, Faculty of Civil Engineeing and Envionmental Engineeing, Depatment of Heat Engineeing,