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.