# Evaporation rate model

Evaporation rate model

In the previous parts, the general equations and the global context have been presented, as well as two examples of application in particular situations :

• pure diffusion
• pure aeration

In this part, the industrial case is studied. This work requires both a theoretical and a numerical study in order to identify the parameters which give the mass concentration profile throughout the container. The aerodynamic equations is intrinsically linked with thermal and mass coefficient. In further, the link between the thermal and the mass mechanism is developed. As referencing with the first part, the Sherwood and the Nusselt number appear in the equations in their dimension form $h_c$ and $h_m$.

1- Mass conservation equation

$$ab\rho v= ab\rho_s v_s (1)$$

2- Spalding mass transport equation

$$\rho v \frac{\partial b}{\partial y}=\frac{\partial}{\partial y}\left(\rho D\frac{\partial b}{\partial y}\right) (2)$$

with the spalding number $b=\frac{\omega}{\omega_S-1}$

3- Heat transport equation

$$\rho v \frac{\partial C_p T}{\partial y}=\frac{\partial}{\partial y}\left(\lambda\frac{\partial T}{\partial y}\right) (3)$$

$$\lambda_s\left(\frac{d T}{d y}\right)_s=h_c(T_\infty-T_s)$$

$h_c(T_\infty-T_s)=\rho_s v_s L_v$

4-  Boundary conditions

The boundary conditions are expressed as follows:

• $b=b_{\infty}$ for $y=\delta_\omega$
• $b=b_{s}$ for $y=h$
• $v_s= D_s \left(\frac{d b}{dy}\right)_s$
• $T=T_\infty$ for $y=\delta_T$
• $T=T_s$ for $y=h$

Therefore, we can simplify the previous expressions some simplifications, the result is :

$W_v(x)=b\rho_sD_s ln(b_{\infty}-b_s+1)\frac{1}{\delta_{\omega}(x)-\delta}$

 $$W_v(x)=\frac{b\lambda}{C_p} ln\left( 1+\frac{W_v(x)}{b}\frac{C_p}{h_c(x)}\right)\frac{1}{\delta_T(x)-\delta}$$

The result is close to the droplet result of J.Reveillon ( Rouen University). The second equation is entirely dependent on thermal conditions. That is why we can have an expression of the mass flux of vapour with the values of the thermal boundary layer $\delta_T$ and the thermal convection coefficient $h_c$. Then we can deduce that the mass flux influences the mass boundary layer.