Problem modeling

Problem modeling

1- Lists of symbols & subscripts

Symbols:

Here are the list of all symbols and subscripts used in the study.

$h$ liquid thickness
$L$ container height dimension
$l_c$ capillary length
$a$ depth
$b$ length
$S_{flux}=a b$ flux surface ($m^2$)
$\rho$ density of the gas mixture ($kg/m^3$)
$P$ pressure ($Pa$)
$u$ the projection of velocity vector along the x-axis ($m/s$)
$\mu$ the dynamic viscosity of the gas mixture ($Pa.s$)
$\nu$ the cinematic viscosity of the gas mixture ($m^2/s$)
$v$ the projection of velocity vector along the y-axis ($m/s$)
$g$ standard gravity ($m/s^2$)
$C_p$ specific heat ($J.kg^{-1}.K^{-1}$)
$T$ temperature of the mixture ($K$)
$\lambda$ the thermal conductivity ($W/m.K$)
$D$ diffusion coefficient of the vapour in the air ($m^2/s$)
$\rho_S$ mass concentration of vapour at the interface ($kg/m^3$)
$\omega$ mass fraction of vapour in the air with $\omega= \frac{\rho_v}{\rho_v+\rho_a}$
$\dot{m}$ mass flow rate ($kg/s$)
$h_{lg}$ latent heat ($kJ/kg$)
$h_c$ thermal exchange coefficient
$h_m$ mass exchange coefficient
$D_h$ hydraulic dimension

Subscript :

2- Assumptions related to the vapour- state :

We consider a layer of liquid in a ventilated box. The following figure represents the problem:

Figure: Problem modeling & Simplifications

During this study, we simplify the problem by considering the following assumptions:

Bidirectional incompressible turbulent flow.
Liquid film thin in relation to the reservoir but sufficient thick for neglect the interfacial resistance.
Air charged in vapour is in thermodynamic balance : the phase change happens in saturation conditions.
Viscous dissipation and pressure work is neglected.
Soret effect ( temperature gradient dependence for the mass flow) and Dufour effect (mass gradient dependence for the heat flux) neglected.
Radiation phenomena neglected.
Boussinesq approximation considered.

In further, the conservation equations are presented below.

Energétique et Procédés