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A theoretical correlation for the Nusselt number in direct contact evaporation of a moving drop in an immiscible liquid

Eine theoretische Beziehung für die Nusselt- Zahl bei Verdunstung eines bewegten Tropfens, der in direktem Kontakt zu einer unmischbaren Flüssigkeit steht

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Abstract

Numerical solutions for the Nusselt number during the direct contact evaporation of a moving drop in a stagnant column of immiscible liquid are presented. The effect of bubble growth rate on the radial component of drop velocity is taken into account in the analysis and the Nusselt number is found to be a function of Peclet number, Jakob number and vapour open angle. A comparison of theoretical and experimental correlations for the Nusselt number shows good agreement. The analysis also yields information on the temperature profile and the thickness of the thermal boundary layer surrounding the evaporating drop.

Zusammenfassung

Es werden numerische Lösungen für die Nusselt-Zahl während der Verdunstung eines bewegten Tropfens, der in direktem Kontakt mit der umgebenden ruhenden Säule aus unmischbarer Flüssigkeit steht, mitgeteilt. In der Berechnung wird der Einfluß der Blasenwachstumsrate auf die radiale Komponente der Tropfengeschwindigkeit berücksichtigt. Es wird festgestellt, daß die Nusselt-Zahl eine Funktion der Peclet-Zahl, der Jakobs-Zahl und des „Öffnungswinkels des Dampfes“ ist. Ein Vergleich der theoretischen und experimentellen Beziehungen für die Nusselt-Zahl zeigt gute Übereinstimmung. Die Berechnung enthält auch Informationen über das Temperaturprofil und die Dicke der thermischen Grenzschicht um den verdampfenden Tropfen.

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Abbreviations

A :

constant in Eq. (4)

B :

diameter ratio

C p :

specific heat of continuous liquid phase

h :

instantaneous heat transfer coefficient

h fg :

latent heat of evaporation of dispersed phase

Ja :

system Jakob number, ϱCpΔ t/(ϱv hfg)

k :

thermal conductivity of continuous liquid phase

m :

mass of liquid fraction in the evaporating drop

m 0 :

total mass of evaporating drop

Nu :

Nusselt number, 2hR/k

Pe :

Peclet number, 2UR/α

Pr :

Prandtl number,ν/α

q :

heat transfer rate per unit surface area of evaporating drop

r :

radial coordinate

R :

instantaneous radius of evaporating drop

Re :

Reynolds number, 2UR/ν

t :

time

T :

temperature

T c :

temperature of continuous liquid phase

T d :

saturation temperature of dispersed phase

U r :

radial component ofU

U Θ :

tangential component ofU

U :

bubble translational velocity

x :

exponent in Eq. (4)

y :

transformed coordinate, (r−R)/R

α :

thermal diffusivity of continuous liquid phase

β :

half vapour open angle

\(\dot \gamma \) :

non-dimensional bubble growth rate,\(\left( {\frac{{dR}}{{dt}} \cdot \frac{R}{\alpha }} \right)\)

ΔT :

temperature difference, (T c −T d )

ϱ :

density of continuous liquid phase

ϱ v :

density of dispersed vapour phase

φ :

non-dimensional temperature,(T− T c )/(T p -T c )

Θ :

spherical polar coordinate

τ :

dimensionless time,α t/R2

μ :

transformed coordinate, (−cosΘ)

ν :

kinematic viscosity of continuous liquid phase

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Authors and Affiliations

  1. Department of Mechanical Engineering, Indian Institute of Technology, 600036, Madras, India

    P. Battya (Research Scholar)

  2. Department of Mechanical Engineering, Indian Institute of Technology, 600036, Madras, India

    V. R. Raghavan (Assistant Professor)

  3. Department of Mechanical Engineering, Indian Institute of Technology, 600036, Madras, India

    K. N. Seetharamu (Professor)

Authors
  1. P. Battya
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  2. V. R. Raghavan
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  3. K. N. Seetharamu
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Battya, P., Raghavan, V.R. & Seetharamu, K.N. A theoretical correlation for the Nusselt number in direct contact evaporation of a moving drop in an immiscible liquid. Warme- und Stoffubertragung 19, 61–66 (1985). https://doi.org/10.1007/BF01682548

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  • DOI: https://doi.org/10.1007/BF01682548

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