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Direct-contact heat transfer to a spherical liquid/vapor two-phase bubble trailing a wake

Wärmeübergang durch Direktkontakt im Nachlauf einer kugelförmigen Zweiphasenblase

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Abstract

The paper deals with the heat transfer causing evaporation of liquid drops in a medium of an immiscible, less volatile liquid. Each drop turns into a two-phase bubble, consisting of a growing vapor phase and a reducing liquid phase, which continues to buoy up in the medium. The bubble is modeled as a sphere in which the yet-to-be vaporized liquid spreads over the rear surface while the rest is occupied by the heat-insulating vapor phase. The rear surface to serve as the effective heat transfer area is assumed to be covered with an axisymmetric wake instead of a boundary layer flow. The quasi-steady, overall heat transfer through the wake in the medium and the layer of the yet-to-be vaporized liquid in the bubble is predicted and compared with relevant experimental results.

Zusammenfassung

Die Untersuchung befaßt sich mit der durch die Verdampfung von Flüssigkeitstropfen in einem aus unvermischbarer, flüchtiger Flüssigkeit bestehendem, Medium verursachten Wärmeübertragung. Aus jedem Tropfen entsteht eine Zweiphasenblase, bestehend aus einer wachsenden Dampfphase und einer abnehmenden Flüssigkeitsphase, die in das Medium aufsteigt. Die Blase wird anhand eines Kugelmodells betrachtet, an dem sich die noch nicht verdampfte Flüssigkeit über die Rückseite verteilt und die restliche Fläche mit der wärmeisolierenden Dampfphase behaftet ist. Es wird angenommen, daß sich an der Rückseite als wirksamer Wärmeübergangsbereich ein achsensymmetrischer Nachlauf bildet. Es werden die quasi-gleichförmige Wärmeübertragung durch den Nachlauf in das Medium und durch die Schicht Lage der nicht verdampften Flüssigkeit in der Blase mathematisch berechnet und mit relevanten Versuchsergebnissen verglichen.

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Abbreviations

D :

diameter of sphere volume-equivalent to bubble

E :

width-to-height aspect ratio of bubble

Eo :

Eötvös number defined in Eq. (1)

g :

acceleration due to gravity

K θ,φ :

local, overall heat transfer coefficient for non-axisymmetricδ distribution

\(\bar K_\theta\) :

azimuthal average ofK θ,φ

m :

exponent of (− cosθ) in Eq. (17)

n :

exponent of sin(φ/2) in Eq. (22)

Nu :

Nusselt number,αD/λ c

Nu r :

Nusselt number related to heat transfer coefficient averaged over the rear half of a sphere

Nu θ :

local Nusselt number,α θ D/λ c

p ∞ :

static pressure in the continuous phase

Pe :

Péclet number,UD/xc

Pr :

Prandtl number,v c/xc

R :

radius of sphere volume-equivalent to bubble

Re :

Reynolds number,UD/ν c

U :

rise velocity of bubble

V :

volume of bubble

V l :

volume of liquid phase in bubble

V v :

volume of vapor phase in bubble

α :

continuous-phase-side heat transfer coefficient based on the surface area of a volume-equivalent sphere

α ov :

overall heat transfer coefficient based on the surface area of a volume-equivalent sphere

α ov :

local, continuous-phase-side heat transfer coefficient

β :

polar angle of three-phase contact line

δ :

local thickness of yet-to-be vaporized liquid on bubble surface

δ b :

δ at the bottom of bubble

δ cr :

critical value of\(\bar \delta \) for the onset of retreating of yet-to-be vaporized liquid on bubble surface

δ θ, max :

the maximum of δ on each parallel of latitude

\(\bar \delta ,\bar \delta _l \) :

average thicknesses of yet-to-be vaporized liquid defined by Eqs. (18) and (27), respectively

\(\bar \delta _\theta\) :

azimuthal average ofδ

Θ :

reduced polar angle defined by Eq. (30)

θ :

polar angle

θ w :

θ at flow-separation line

x:

thermal diffusivity

λ :

thermal conductivity

ν :

kinematic viscosity

ζ :

mass fraction of vapor in bubble

ϱ :

mass density

σ :

‘vapor/continuous-phase liquid’ interfacial tension

φ :

azimuthal angle

ψ :

constant indicating the ratio of (\(\bar \delta _l - \bar \delta \)) to (\(\bar \delta _{cr} - \bar \delta \))

c :

continuous-phase liquid

dl :

dispersed-phase substance in the state of saturated liquid

dν :

dispersed-phase substance in the state of saturated vapor

0:

initial condition before the onset of evaporation

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Authors and Affiliations

  1. Department of Mechanical Engineering Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, 223, Yokohama, Japan

    Y. H. Mori & N. Ehara

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  1. Y. H. Mori
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  2. N. Ehara
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Mori, Y.H., Ehara, N. Direct-contact heat transfer to a spherical liquid/vapor two-phase bubble trailing a wake. Wärme- und Stoffübertragung 27, 337–345 (1992). https://doi.org/10.1007/BF01600023

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