Equations

Equations                                                


 

       1- Non-dimensional equations

For a greatest lisibility of the mechanisms, the non dimensional equation can be written as :

$$X=\frac{x}{L} , Y=\frac{y}{L} , U=\frac{u}{U_{in}} , V=\frac{v}{V_{in}}, D_{hydraulic}=L$$ ,

$$ \theta=\frac{T-T_{in}}{T_s-T_{in}} ,  C=\frac{\omega-\omega_{in}}{\omega_s-\omega_{in}}$$

The non dimensional equations become :

$$U\frac{\partial U}{\partial X}+V\frac{\partial U}{\partial Y}=\frac{1}{Re}\left(\frac{\partial^{2}U}{\partial X^{2}}+\frac{\partial^{2}U}{\partial Y^{2}}\right)    (1*)$$

$$U\frac{\partial V}{\partial X}+V\frac{\partial V}{\partial Y}=-\frac{\partial P_{m}}{\partial Y}+\frac{1}{Re}\left(\frac{\partial^{2}V}{\partial X^{2}}\right)      (2*)$$

$$U\frac{\partial \theta}{\partial X}+V\frac{\partial \theta}{\partial Y}=\frac{1}{RePr}\left(\frac{\partial^{2}\theta}{\partial X^{2}}+\frac{\partial^{2}\theta}{\partial Y^{2}}\right)    (3*)$$

$$U\frac{\partial C}{\partial X}+V\frac{\partial C}{\partial Y}=\frac{1}{ReSc}\left(\frac{\partial^{2}C}{\partial X^{2}}+\frac{\partial^{2}C}{\partial Y^{2}}\right)      (4*)$$

 

        2- Non-dimensional numbers

Some interesting numbers appear in this non-dimensional equation.

  • Reynolds number

The Reynolds number is characteristic of pure flow conditions, inertial effect compared with momentum diffusion:

$$Re_{in}=\frac{\rho_{in}LU_{in}}{\mu}$$

  • Prandlt number

Another one is the Prandlt number. It appears in thermal equation and it compares the momentum diffusion with the thermal diffusion

$$Pr=\frac{\frac{\mu}{\rho_{in}}}{\frac{\lambda}{\rho C_p}}$$

  • Schimdt number

The Schimdt number, quiet close with the Prandt number but it appear in mass equation. It compares the movement quantity diffusion with the molecular diffusivity.

$$Sc=\frac{\frac{\mu}{\rho_{in}}}{D}$$

In addition the expression of the thermal flows through the liquid film has to be written clearly because the study of the heat transfers can explain the energy transfer between the liquid phase and the vapour phase.

Firstly, the sensible heat flux can be modeled as:

$$ q''_s=-\lambda_{liq} \left(\frac{\partial T}{\partial y}\right)_{y=h}          (5) $$

That is the energy transfer through the thickness of the liquid  resulting from temperature gradient. We can suppose in this case the convection is neglected.

 

Then the local energy balance on the interface is :

$$q''_{evap}=q''_{fluid}+q''_ {gas}$$

that is to say :

$$h_{lg}h_m(\rho_{in}-\rho_{S})=h_c(T_{in}-T_S)+\lambda_l \left(\frac{\partial T}{\partial y}\right)_{y=h}$$

  • Nusselt number & Sherwood number

After all that, the Nusselt number and the Sherwood number are interesting to be studied. These numbers compare the convective effect and the diffusion effect.

The local Nusselt number compares the thermal convection flux with the thermal diffusion flux and it is defined as :

$$ Nu=\left(\frac{\partial \theta}{\partial Y}\right)_{Y=h/L}$$

Then we can deduce the global Nusselt Number as :

$$\bar{Nu}=\frac{\bar{h_c}*D_h}{k}=\frac{q_t*D_h}{\lambda(T_{in}-T_{S})}$$

 

To finish this first part, the Sherwood number compares mass convective flux with molecular diffusion.

$Sh=\frac{h_m*D_h}{D}$ with $h_m=\frac{\dot{m}}{\rho(\omega_{in}-\omega_{S})}$

The total flux of vapour mass is $\dot{m}=\rho_l v_l=\rho_S v_S=\omega_S\rho v_S-\rho D(\frac{\partial \omega}{\partial y})_{y=h}$.

 

     N.B:

This work was made according to the thesis "Transfert couplé de chaleur et de masse par convection mixte avec changement de phase dans un canal" fulfilled by Othmane OULAID.