# Falling Film Evaporation on a Heated Wall

In falling film evaporators, the flowing gas provides the latent heat of vaporization, and the gas cools as it passes through the evaporator. If the gas flows downward, it grows denser as it cools. The buoyancy force creates a chimney effect, by means of which the gas exits the bottom and more gas enters the top. An experimental study of this flow was conducted by Yan et al. (1991) and followed by a numerical simulation by Yan and Lin (1991), which is presented below. The figure on the right shows the physical model of evaporation from a vertical falling film on two parallel walls with a spacing of 2b. While the outer sides of the wall are insulated, the liquid falling films on the inner walls evaporate into a gas that flows through the channel. Assuming there is no pressure gradient and the inertial term is negligible, the momentum balance in the assumed laminar liquid film with negligible inertia is $\frac{\partial }{\partial y}\left[ \left( {{\mu }_{\ell }}+{{{{\mu }'}}_{\ell }} \right)\frac{\partial {{u}_{\ell }}}{\partial y} \right]+{{\rho }_{\ell }}g=0$ (1)

where ${{{\mu }'}_{\ell }}$ is a correction to the viscosity in response to presumed surface waves on the film. Considering conduction through the liquid and sensible heat loss, the energy balance is ${{\rho }_{\ell }}{{c}_{p,\ell }}{{u}_{\ell }}\frac{\partial {{T}_{\ell }}}{\partial x}=\frac{\partial }{\partial y}\left[ \left( {{k}_{\ell }}+{{{{k}'}}_{\ell }} \right)\frac{\partial {{T}_{\ell }}}{\partial y} \right]$ (2)

where ${{{k}'}_{\ell }}$ is a correction to the conductivity to account for the effect of the surface waves. Natural convection occurs in the gas region, and the continuity equation is $\frac{\partial }{\partial x}{{\left( \rho u \right)}_{g}}+\frac{\partial }{\partial y}{{\left( \rho v \right)}_{g}}=0$ (3)

The momentum equation in the gas region is considered only in the axial direction, and constant density over the cross-section is assumed ${{(\rho u)}_{g}}\frac{\partial {{u}_{g}}}{\partial x}+{{(\rho v)}_{g}}\frac{\partial {{u}_{g}}}{\partial y}=-\frac{d{{p}_{dyn}}}{dx}+\frac{\partial }{\partial y}\left[ \left( {{\mu }_{g}}+{{{{\mu }'}}_{g}} \right)\frac{\partial {{u}_{g}}}{\partial y} \right]+\left( {{\rho }_{g}}-{{\rho }_{0}} \right)g$ (4)

where pdyn is dynamic pressure defined as ${{p}_{dyn}}=p-\frac{1}{2}\rho u_{0}^{2}$ (5)

The energy equation includes energy storage by the gas, conduction through the gas, and convection of the vapor component through the gas. \begin{align} & {{(\rho {{c}_{p}}u)}_{g}}\frac{\partial {{T}_{g}}}{\partial x}+{{(\rho {{c}_{p}}v)}_{g}}\frac{\partial {{T}_{g}}}{\partial y} \\ & =\frac{\partial }{\partial y}\left[ \left( {{k}_{g}}+{{{{k}'}}_{g}} \right)\frac{\partial {{T}_{g}}}{\partial y} \right]+{{\rho }_{g}}\left( {{\alpha }_{g}}+{{{{\alpha }'}}_{g}} \right)\left( {{c}_{p,v}}-{{c}_{p,g}} \right)\frac{\partial {{T}_{g}}}{\partial y}\frac{\partial \omega }{\partial y} \\ \end{align} (6)

where the diffusivity is corrected to reflect the effect of surface waves. The last term on the right-hand side of eq. (6) represent the contribution of mass concentration gradient on the energy balance.

A concentration balance is also required: ${{(\rho u)}_{g}}\frac{\partial \omega }{\partial x}+{{(\rho v)}_{g}}\frac{\partial \omega }{\partial y}=\frac{\partial }{\partial y}\left[ {{\rho }_{g}}\left( D+{D}' \right)\frac{\partial \omega }{\partial y} \right]$ (7)

The boundary conditions at the inlet are now specified with uniform temperature, velocity, and concentration: $x=0\begin{matrix} : & u={{u}_{0}} & T={{T}_{0}} & \begin{matrix} \omega ={{\omega }_{0}} & {{p}_{dyn}}=-\rho u_{0}^{2}/2 \\ \end{matrix} \\ \end{matrix}$ (8)

At the center of the channel, symmetric conditions must be satisfied, i.e., $y=0:\quad \frac{\partial u}{\partial y}=0,\quad \frac{\partial T}{\partial y}=0,\quad \frac{\partial \omega }{\partial y}=0$ (9)

At the gas-liquid interface, the velocity, temperature, and concentration are at their interface values: $y=b-\delta :\quad u={{u}_{\delta }},\quad T={{T}_{\delta }},\quad \omega ={{\omega }_{\delta }}$ (10)

With the nonslip condition, the shear stresses act equally and in opposite directions: ${{\tau }_{\delta }}={{\left[ \left( \mu +{\mu }' \right)\frac{\partial u}{\partial y} \right]}_{\ell }}={{\left[ \left( \mu +{\mu }' \right)\frac{\partial u}{\partial y} \right]}_{g}}$ (11)

The mass balance that accounts for the mass flux convected from the interface is ${\dot{m}}''=\frac{\rho \left( D+{D}' \right)}{1-{{\omega }_{\delta }}}{{\left( \frac{\partial \omega }{\partial y} \right)}_{\delta }}$ (12)

An energy balance that accounts for the heat required to release the mass from the liquid film is ${{\left[ \left( k+{k}' \right)\frac{\partial T}{\partial y} \right]}_{\ell }}={{\left[ \left( k+{k}' \right)\frac{\partial T}{\partial y} \right]}_{g}}+{\dot{m}}''{{h}_{\ell v}}$ (13)

The mass fraction at the interface is assumed to be that for thermodynamic equilibrium $\omega_{\delta} = \frac{M_{v}p_{\delta}}{M_g (p-p_{\delta})+M_v p_{\delta}}$ (14)

where Mv and Mg are the molecular masses of vapor and gas, respectively. At the channel wall there is also a nonslip condition with the liquid, and the wall is adiabatic – $y=b: u_l =0, \frac{\partial T_l}{\partial y} = 0$ (15)

At the channel exit the dynamic pressure falls to zero:

 x = 0:pdyn = 0 (16)

Corrections to the viscosity due to presumed surface waves on the film are (Yan and Lin, 1991): ${{{\mu }'}_{\ell }}={{c}_{\ell }}{{\rho }_{\ell }}{{{\delta }'}^{2}}f{{d}_{\ell }}$ (17)
 μ'g = cgρgδ'2fdg (18)

where the values of cl and cg are 0.1 and 10, respectively. The amplitude, δ'(m), and frequency, f(Hz), of the wave are estimated by the following empirical correlations: ${\delta }'=\left\{ \begin{matrix} 4.1236\times {{10}^{-4}}\text{W}{{\text{e}}^{0.5505}} \\ 1.5340\times {{10}^{-4}}\text{W}{{\text{e}}^{0.2472}} \\ \end{matrix} \right.\begin{matrix} {} & \begin{matrix} \text{We}\le 0.038357 \\ \text{We}>0.038357 \\ \end{matrix} \\ \end{matrix}$ (19) $f=\left\{ \begin{matrix} 86.70\text{W}{{\text{e}}^{0.371327}} \\ 71.79\text{W}{{\text{e}}^{0.319413}} \\ \end{matrix} \right.\begin{matrix} {} & \begin{matrix} \text{We}\le 0.0264443 \\ \text{We}>0.0264443 \\ \end{matrix} \\ \end{matrix}$ (20)

where We is Weber number, defined as $\text{We}=\frac{{{\left( {{\operatorname{Re}}_{\delta }}/4 \right)}^{5/3}}}{\Omega }$ (21)

and the Reynolds number, ${{\operatorname{Re}}_{\delta }}$, and surface tension parameter, Ω, are ${{\operatorname{Re}}_{\delta }}=\frac{4\Gamma }{{{\mu }_{\ell }}}$ (22) $\Omega =0.6736219\frac{\sigma }{{{\rho }_{\ell }}\nu _{\ell }^{4/3}}$ (23)

The damping functions ${{d}_{\ell }}$ and dg in eqs. (17) and (18) are obtained by ${{d}_{\ell }}=1-\exp \left( -\frac{b-y}{\delta } \right)$ (24) ${{d}_{g}}=\exp \left[ -\frac{5(b-\delta -y)}{\delta } \right]$ (25)

The correction on thermal diffusivity, α', can be estimated by introducing the wave Prandtl number, $\operatorname{P}{r}'={\nu }'/{\alpha }'$. Similarly, the correction on mass diffusivity is estimated by introducing the wave Schmidt number, Sc' = D' / α'. It is recommended that the values of wave Prandtl and Schmidt numbers be unity and that the problem be solved numerically.

Experimental results using ethanol revealed that a larger inlet film temperature increases the liquid film cooling; this was to be expected because the temperature differential between the film and gas was increased. Lowering the liquid mass flow rate also increases the rates liquid film cooling; this is most likely due to longer residence times in the channel. It was also found that the exit temperature of the film could drop below the ambient temperature, which can be expected due to the volatility of ethanol.

Yan et al. (1991) performed an experimental study of evaporative cooling of falling liquid film in an adiabatic vertical channel. Their experimental data yielded an empirical correlation for the mean heat flux, $\bar{{q}''},$ of water film evaporation: $\bar{{q}''}=0.0113T_{\ell ,0}^{3.15}{{b}^{{1}/{3}\;}}$ (26)

where ${{T}_{\ell ,0}}$ is the liquid film inlet temperature. It can be seen from eq. (26) that the heat flux is not a function of the mass flow rate of the liquid film.

## References

Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Elsevier, Burlington, MA

Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.

Yan, W.M., Lin, T.F., and Tsay, Y.L., 1991, “Evaporative Cooling of a Liquid Film through Interfacial Heat and Mass Transfer in a Vertical Channel – I. Experimental Study,” International Journal of Heat and Mass Transfer, Vol. 34, pp. 1105-1111.

Yan, W.M., and Lin, T.F., 1991, “Evaporative Cooling of a Liquid Film through Interfacial Heat and Mass Transfer in a Vertical Channel – II. Numerical Study,” International Journal of Heat and Mass Transfer, Vol. 34, pp. 1113-1124.