Abstract
Under microgravity conditions, the dynamics of a thin condensate film on a curved surface is determined by the capillary pressure gradient proportional to the mean surface curvature gradient. A one-parameter family of axisymmetric surfaces is found for which the gradient of mean curvature is constant. The dimensionless equation for rotation angle of the generatrix curve is found. There is a single generatrix curve for an axisymmetric surface for which the rotation angle at the inflexion point assumes a predetermined value. Due to the constant gradient of capillary pressure on such a surface, a stable condensate flow is ensured under microgravity conditions. A similar curve for the planar case, known as the clothoid or "Cornu spiral", is used to find the best transition curve to get the smoothest traffic on the roads. A numerical model for film-wise vapor condensation on such surface has been built. The film thickness distribution and mass flow rate of the HFE-7100 along the cooled curvilinear fin have been calculated. Calculations were done both for terrestrial gravity and microgravity. This work proposes a particular surface shape, found numerically, for conducting experiments on the pure vapor condensation under microgravity conditions in Parabolic Flight Campaigns and onboard the International Space Station.
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Data Availability
The data generated during and analyzed during the current study are available from the corresponding author on reasonable request.
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Abbreviations
- P :
-
capillary pressure, N/m2
- \(\frac{\partial P}{\partial S}\) :
-
capillary pressure gradient along the generatrix curve, N/m3
- σ :
-
surface tension coefficient, N/m
- κ(s) :
-
curvature of the generatrix curve, \({m}^{-1}\)
- κ 0 :
-
curvature on the fin top, \({m}^{-1}\)
- \({\kappa }_{1}{,\kappa }_{2}\) :
-
main curvatures, \({m}^{-1}\)
- \({\kappa }_{fin}=\kappa +sin\left(\theta \right)/r\) :
-
double mean curvature of fin surface, \({m}^{-1}\)
- s :
-
variable along the curved surface S
- \({S}_{fin}\) :
-
end of the fin's curve
- \(r\left(s\right)=\underset{0}{\overset{s}{\int }}cos\left(\theta \left(\xi \right)\right)d\xi\) :
-
radius of the fin along s
- \(y\left(s\right)=\underset{0}{\overset{s}{\int }}cos\left(\theta \left(\xi \right)\right)d\xi\) :
-
coordinate, directed downward
- \(\theta \left(s\right)=\underset{0}{\overset{s}{\int }}\kappa \left(\xi \right)d\xi\) :
-
rotation angle of the curve at s
- \(b=a/{\kappa }_{0}^{2}\) :
-
curve's parameter
- \({V}_{\Omega }=\underset{\Omega }{\overset{}{\int }}hds\) :
-
liquid volume
- Ω :
-
is an element of the surface S
- \(J=-\lambda\Delta T/{\rho r}_{lv}h\) :
-
is volumetric flux of the condensate through the interface
- h t :
-
is velocity of the condensate film, m/s
- λ :
-
is thermal conductivity of liquid, W/m/K
- r lv :
-
is latent heat of vaporization, J/kg
- ρ :
-
is density, kg/m3
- \(\overrightarrow q=\frac{h^3}{3\mu}\overrightarrow f+\frac{h^2}{2\mu}{\overrightarrow\tau}_{sur}\) :
-
is the liquid flow vector along the surface, m2/s
- \(\overrightarrow{f}\) :
-
gradient of the modified pressure, Pa/m
- μ :
-
is dynamic viscosity, kg/m/s
- h :
-
liquid film thickness, m
- h 0 :
-
initial liquid film thickness, m
- t :
-
time, s
- ΔT = Tsat-Tw :
-
temperature drop, K
- Tw :
-
temperature of the fin's surface, oC
- Tsat :
-
saturation temperature, oC
- Bo :
-
the Bond number
- ESA :
-
European Space Agence
- ISS :
-
International Space Station
- PFC :
-
Parabolic Flight Campaign
- IP :
-
inflexion point
- HFE-7100 :
-
trademark of methoxy-nanouorobutane (C4F9OCH3)
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This work was carried out under state contract with IT SB RAS (121031200084-2)
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This work was carried out under state contract with IT SB RAS (121031200084–2).
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Barakhovskaia, E., Marchuk, I. Fin Shape Design for Stable Film-Wise Vapor Condensation in Microgravity. Microgravity Sci. Technol. 34, 8 (2022). https://doi.org/10.1007/s12217-021-09918-z
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DOI: https://doi.org/10.1007/s12217-021-09918-z