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 :
- $v$ for the vapour
- $S$ for the vapour at the interface
- $in$ for the input condition
- $w$ for the wall condition
- $l$ for the liquid
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.