Molecular gas dynamics applied to phase change processes at a vapor–liquid interface: shock-tube experiment and MGD computation for methanol

Abstract

This paper deals with a molecular gas-dynamics method applied to the accurate determination of the condensation coefficient of methanol vapor. The method consisted of an experiment using a shock tube and computations using a molecular gas-dynamics equation. The experiments were performed in such situations where the shift from a vapor–liquid equilibrium state to a nonequilibrium one is realized by a shock wave in a scale of molecular mean free time of vapor molecules. The temporal evolution in thickness of a liquid film formed on the shock-tube endwall behind a reflected shock wave is measured by an optical interferometer. By comparing the measured liquid-film thickness with numerical solutions for a polyatomic version of the Gaussian–BGK model of the Boltzmann equation, the condensation coefficient of methanol vapor is accurately determined in vapor–liquid nonequilibrium states. As a result, it is clear that the condensation coefficient is just unity very near to an equilibrium state, but is smaller far from the equilibrium state.

Appendix

The mathematical formulation of the problem has already been shown in our recent paper (Yano et al. 2003). We present here the governing equation, boundary condition, and main parameters used in the calculations for readers’ convenience.

The present problem is a spatially one-dimensional formulation. The polyatomic version of the Gaussian–BGK model of Boltzmann equation can therefore be written as (Andries et al. 2000)

$$ \frac{{\partial f}} {{\partial t}} + {\left( {\xi _{1} - u_{{\ell }} } \right)}\frac{{\partial f}} {{\partial x}} = \frac{p} {{\mu {\left( {1 - \nu + \theta \nu } \right)}}}{\left[ {G{\left( f \right)} - f} \right]}, $$

(4)

where f is the velocity distribution function for methanol vapor molecules, x is the spatial coordinate measured from the surface of the growing liquid film, ξ ₁(=ξ _x ) is the x component of molecular velocity, u _ℓ is the growing speed of the liquid film, p is the vapor pressure, µ is the viscosity, and θ and ν are parameters (see below). The right-hand side represents the effect of molecular collisions and G(f) is defined as

$$ G{\left( f \right)} = \rho a{\left( f \right)}b{\left( f \right)}, $$

(5)

$$ a{\left( f \right)} = \frac{1} {{{\sqrt {\det {\left( {2\pi \tau _{{ij}} } \right)}} }}}\exp {\left[ { - \frac{1} {2}{\left( {\xi _{i} - u_{i} } \right)}\tau ^{{ - 1}}_{{ij}} {\left( {\xi _{j} - u_{j} } \right)}} \right]}, $$

(6)

$$ b{\left( f \right)} = \frac{1} {{\Gamma {\left( {\frac{n} {2} + 1} \right)}{\left( {RT_{{{\text{rel}}}} } \right)}^{{n/2}} }}\exp {\left( { - \frac{{\eta ^{{2/n}} }} {{RT_{{{\text{rel}}}} }}} \right)}, $$

(7)

where ρ is the vapor density, u _i the fluid velocity, τ _ij a rectified stress tensor, T _rel the relaxation temperature, Γ the gamma function, det the determinant, and η ^2/n the rotational energy of a molecule with n degrees of freedom of rotation. The definitions of macroscopic variables in Eqs. 5, 6, and 7, ρ, u _i , τ _ij , and T _rel, are given through the following equations:

$$ \begin{array}{*{20}l} {{\rho = {\int {f{\text{d}}{\mathbf{\xi }}{\text{d}}\eta ,} }} \hfill} & {{u_{i} = \delta _{{i1}} u,} \hfill} & {{\rho u = {\int {\xi _{1} f{\text{d}}{\mathbf{\xi }}{\text{d}}\eta } }} \hfill} \\ \end{array} , $$

(8)

$$ 3\rho RT_{{{\text{tr}}}} = {\int {{\left( {\xi _{i} - u_{i} } \right)}^{2} f{\text{d}}{\mathbf{\xi }}{\text{d}}\eta } }, $$

(9)

$$ n\rho RT_{{\operatorname{int} }} = 2{\int {\eta ^{{2/n}} f{\text{d}}{\mathbf{\xi }}{\text{d}}\eta } }, $$

(10)

$$ {\left( {3 + n} \right)}RT = 3RT_{{{\text{tr}}}} + nRT_{{\operatorname{int} }} , $$

(11)

$$ \begin{array}{*{20}l} {{p = \rho RT,} \hfill} & {{T_{{{\text{rel}}}} = \theta T} \hfill} \\ \end{array} + {\left( {1 - \theta } \right)}T_{{\operatorname{int} }} , $$

(12)

$$ \rho \Theta _{{ij}} = {\int {{\left( {\xi _{i} - u_{i} } \right)}{\left( {\xi _{j} - u_{j} } \right)}f{\text{d}}{\mathbf{\xi }}{\text{d}}\eta } }, $$

(13)

$$ \tau _{{ij}} = {\left( {1 - \theta } \right)}{\left[ {{\left( {1 - \nu } \right)}RT_{{{\text{tr}}}} \delta _{{ij}} + \nu \Theta _{{ij}} } \right]} + \theta RT\delta _{{ij}} , $$

(14)

where ξ=(ξ ₁,ξ ₂,ξ ₃) is the molecular velocity, T _tr and T _int are temperatures associated with the translational and rotational motions of molecules, respectively, T is the vapor temperature, Θ _ij the stress tensor, and δ _ij Kronecker’s delta. The three-dimensional integration with respect to ξ is performed over the whole space of ξ, and the integration with respect to η is made over 0≤η<∞.

The most important feature of the Gaussian–BGK model is that it can give a realistic Prandtl number and second viscosity coefficient, by adjusting the parameters θ and ν in Eqs. 4, 12, and 14 (Andries et al. 2000). For the methanol vapor, we set θ=11/17 and ν=−1/2, and this gives the Prandtl number of 0.85 and the ratio of the first and second viscosity coefficients 0.8 (for air, Lighthill 1956). The parameter n is set as n=6 to approximate an experimental value of the ratio of specific heats of methanol vapor, γ=1.228 (290 K).

The boundary condition at the vapor–liquid interface is given by

$$ f{\left( {x = 0,\xi _{1} > u_{{\ell }} } \right)} = \alpha f^{*} + {\left( {1 - \alpha } \right)}f^{r} , $$

(15)

$$ \begin{array}{*{20}l} {{f* = \rho *a_{{\text{M}}} b_{{\text{M}}} ,} \hfill} &amp; {{f^{r} = \sigma _{{\text{w}}} a_{{\text{M}}} b_{{\text{M}}} } \hfill} \\ \end{array} , $$

(16)

$$ a_{{\text{M}}} = \frac{1} {{{\left( {2\pi RT_{{\ell }} } \right)}^{{3/2}} }}\exp {\left[ { - \frac{{{\left( {\xi _{1} - u_{{\ell }} } \right)}^{2} + \xi ^{2}_{2} + \xi ^{2}_{3} }} {{2RT_{{\ell }} }}} \right]}, $$

(17)

$$ b_{{\text{M}}} = \frac{1} {{\Gamma {\left( {\frac{n} {2} + 1} \right)}{\left( {RT_{{\ell }} } \right)}^{{n/2}} }}\exp {\left( { - \frac{{\eta ^{{2/n}} }} {{RT_{{\ell }} }}} \right)}, $$

(18)

where ρ*=p*/(RT _ℓ ) is the saturated vapor density and σ _w is determined from the velocity distribution function of molecules coming from the vapor, f=f ⁱ (Sone 2002)

$$ \sigma _{{\text{w}}} = - {\sqrt {\frac{{2\pi }} {{RT_{{\ell }} }}} }{\int_{\xi _{1} &lt; u_{l} } {{\left( {\xi _{1} - u_{{\ell }} } \right)}f^{i} {\text{d}}{\mathbf{\xi }}{\text{d}}\eta } }. $$

(19)

The saturated vapor pressure p* is evaluated from Antoine’s formula, which can be written for methanol vapor as

$$ \begin{array}{*{20}l} {{p* = \exp {\left( {A - \frac{B} {{C + T_{{\ell }} }}} \right)}} \hfill} &amp; {{{\left( {{\text{Pa}}} \right)},} \hfill} \\ \end{array} $$

(20)

where A=23.4803, B=3,626.55, and C=−34.29 (Japanese Society of Chemical Engineers 1985).