Evaporation Rates and Bénard-Marangoni Supercriticality Levels for Liquid Layers Under an Inert Gas Flow

Abstract

In this work, we propose an approximate model of evaporation-induced Bénard-Marangoni instabilities in a volatile liquid layer with a free surface along which an inert gas flow is externally imposed. This setting corresponds to the configuration foreseen for the ESA—“EVAPORATION PATTERNS” space experiment, which involves HFE-7100 and nitrogen as working fluids. The approximate model consists in replacing the actual flowing gas layer by an “equivalent” gas at rest, with a thickness that is determined in order to yield comparable global evaporation rates. This allows studying the actual system in terms of an equivalent Pearson’s problem (with a suitably defined wavenumber-dependent Biot number at the free surface), allowing to estimate how far above critical the system is for given control parameters. Among these, a parametric analysis is carried out as a function of the liquid-layer thickness, the flow rate of the gas, its relative humidity at the inlet, and the ambient pressure and temperature.

Appendix: Relevant Fluid Properties

This section gives the material property values required in the present study.

Pure-Fluid Properties

The values tabulated below are taken partly from 3M^TM Novec^TMHFE-7100 and HFE-7300 Engineered Fluid data sheets (3M 2012) and Gas Encyclopedia (Airliquide 2012) (the latter for N ₂), and partly from private communications within the ESA Topical Team (see Table 8).

Table 8 Properties of the pure fluids involved (at 1 atm, when relevant)

For the saturation pressure as a function of temperature, we use

$$ p_{sat}(T)=p_{sat} ({T_{0}})\text{exp}\left( {-\frac{L_{0} M}{R}\left( {\frac{1}{T}-\frac{1}{T_{0} }} \right)} \right) $$

where M is the molecular mass, $R\approx 8,31\frac {\text {J}}{\text {mol\;K}}$ is the universal gas constant, T ₀ is some reference temperature, and L ₀is the latent heat of evaporation at this temperature. At other temperatures, L is evaluated according to L = L ₀ + (c _pl − c _pg1)(T ₀ − T).

Gas Mixture Properties

The following ones are explicitly needed in the present study: ρ _g (density), λ _g (thermal conductivity), c _pg (specific heat, here per unit mass) and D _g (diffusion coefficient). The gas phase is a mixture of two pure components “1” and “2” (here HFE vapor and N ₂, respectively), and the corresponding supplementary subscripts will refer to the corresponding pure-substance properties (e.g. λ _g1, c _pg2, M ₁ , etc.). Let x _g and N _g denote the molar and mass fractions of the first substance (here HFE) in the gas mixture, easily expressible through one another:

$$\begin{array}{@{}rcl@{}} x_{g} &=&\frac{N_{g}}{N_{g} +({1-N_{g}}){M_{1}}/{M_{2} }}, \notag \\ N_{g} &=&\frac{x_{g} }{x_{g} +( {1-x_{g} }){M_{2}}/{M_{1}}} \end{array} $$

Density, $\rho _{g}$

Using the perfect-gas laws, one obtains

$$\rho_{g} ( {x_{g} ,T,p})=\frac{p}{RT}( {( {M_{1} -M_{2} })x_{g} +M_{2} }) $$

Specific Heat, $c_{pg}$

In the perfect-gas approximation, we have

$$c_{pg} =c_{pg1} N_{g} +c_{pg2} ({1-N_{g}}) $$

which can subsequently be expressed in terms of x _g , if needed.

Thermal Conductivity, $\lambda _{g}$

It is evaluated according to Wassiljewa (1904) formula. At moderate pressure (less or equal to 1 atm), the latter reads

$$\lambda_{g} ({x_{g} ,T})=\frac{\lambda_{g1}(T)}{1+\frac{1-x_{g} }{x_{g} }\varphi_{12} ( T )}+\frac{\lambda_{g2} (T)}{1+\frac{x_{g} }{1-x_{g} }\varphi_{21} (T)} $$

Several correlations are currently available in the literature that differ by the method defining the functions φ ₁₂ and φ ₂₁. In the present study, we stick to the Lindsay and Bromley (1950) method, as used by Perry and Green (1997). Assuming the case of nonpolar molecules, one then has

$$\begin{array}{@{}rcl@{}} \varphi_{12} (T)&=&\frac{1}{4}\left( {1+\sqrt{\frac{\mu_{g1} ( T )}{\mu_{g2} ( T)}\left( {\frac{M_{2} }{M_{1} }} \right)^{3/4}\frac{T+1.5T_{b1} }{T+1.5T_{b2} }}} \right)^{2} \notag \\ &&\qquad\times\frac{T+1.5\sqrt{T_{b1} T_{b2} }}{T+1.5T_{b1} }, \notag \\ [12pt] \varphi_{21} (T)&=&\frac{1}{4}\left( {1+\sqrt{\frac{\mu_{g2} ( T )}{\mu_{g1} ( T )}\left( {\frac{M_{1} }{M_{2} }} \right)^{3/4}\frac{T+1.5T_{b2} }{T+1.5T_{b1} }}} \right)^{2} \notag \\ &&\qquad\times\frac{T+1.5\sqrt{T_{b1} T_{b2} }}{T+1.5T_{b2} } \end{array} $$

Here μ _g is the dynamic viscosity. The subscript b refers to the boiling point of the corresponding component at 1 atm. The study by Tondon and Saxena (1968) (where 96 mixtures are considered) thereby reproduces the experimental data with an average deviation of 2.20 %. It is assumed that the gas dynamic viscosity and thermal conductivity do not depend on the pressure.

Diffusion Coefficient, $D_{g}$

At moderate pressures (less or equal to 1 atm), the diffusivity of a gas binary mixture can be deduced from the relation due to Slattery and Bird (1958) (see also Bird et al. 2007, page 521). The result is independent of the system composition, varies as the inverse of the pressure and approximately as the square of the temperature. For nonpolar gas pairs, one has

$$\begin{array}{@{}rcl@{}} D_g ({T,p})&=&2.745 \times 10^{-4}\frac{1}{p} \left( {\frac{1}{M_1 }+\frac{1}{M_2 }} \right)^{{1}/{2}}\left( {\frac{T}{\sqrt{T_{cr1} T_{cr2} }}} \right)^{1.823}\notag\\[8pt]&&\times( {T_{cr1} T_{cr2} } )^{{5}/{12}} ( {p_{cr1} p_{cr2} } )^{{1}/{3}} \end{array} $$

(12)

Here the molar weight must be in g/mol, the pressure in atm, the diffusion coefficient in cm ²/s, and the temperature in K. The subscript “cr” refers to the critical properties of the mixture components. Table 9 shows the values of D _g for cases 1–5 calculated using this formula.

Table 9 The values of the diffusion coefficient in the gas for cases 1–5