Turbulent falling film evaporation

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Stephan (1992) presented an empirical relationship between Reynolds number and Prandtl number that could be used to determine whether a falling film is completely turbulent:

\operatorname{Re}\ge {{\operatorname{Re}}_{turb}}=5840{{\Pr }^{-1.05}} (9.156)

In order to model turbulent flow, the following development from Seban and Faghri (1976) proceeds with assumptions similar to those made in the classical Nusselt laminar analysis. The momentum equation is

\frac{d}{dy}\left( {{v}_{\ell }}+{{\varepsilon }_{M}} \right)\frac{du}{dy}+g=0 (9.157)

where {{\varepsilon }_{M}} is the turbulent eddy diffusivity.

Defining the following dimensionless variables,

{{u}_{\tau }}=\sqrt{{{{\tau }_{w}}}/{{{\rho }_{\ell }}}\;},\ \text{ }{{u}^{+}}={u}/{{{u}_{\tau }},\ \ {{y}^{+}}={y{{u}_{\tau }}}/{{{v}_{\ell }}}\;}\; (9.158)

the momentum equation becomes

\frac{d}{d{{y}^{+}}}\left( \frac{{{\varepsilon }_{M}}}{{{v}_{\ell }}}+1 \right)\frac{d{{u}^{+}}}{d{{y}^{+}}}+\frac{g{{v}_{\ell }}}{u_{\tau }^{3}}=0 (9.159)

which is subject to the following boundary conditions:

   , & {{y}^{+}}=0  \\
\end{matrix} (9.160)
   , & {{y}^{+}}=\delta   \\
\end{matrix}}^{+}} (9.161)

Integrating eq. (9.159) twice and using eqs. (9.160) and (9.161) to determine the integral constant, the dimensionless velocity becomes

{{u}^{+}}=\int_{0}^{{{y}^{+}}}{\frac{\left( 1-{{{y}^{+}}}/{{{\delta }^{+}}}\; \right)}{\left( 1+{{{\varepsilon }_{M}}}/{{{v}_{\ell }}}\; \right)}d{{y}^{+}}} (9.162)

The energy equation is

u\frac{\partial T}{\partial x}=\frac{\partial }{\partial y}\left( {{\varepsilon }_{H}}+\alpha  \right)\frac{\partial T}{\partial y} (9.163)

where {{\varepsilon }_{H}} is turbulent eddy thermal diffusivity. Equation (9.163) is subject to the following boundary conditions:

   T={{T}_{sat}} & x=0  \\
\end{matrix} (9.164)
\frac{\partial T}{\partial y}=-\frac{{{{{q}''}}_{w}}}{{{k}_{\ell }}}\begin{matrix}
   , & y=0  \\
\end{matrix} (9.165)
   , & y=\delta   \\
\end{matrix} (9.166)

By defining dimensionless temperature as

{{T}^{+}}=\frac{T{{\rho }_{\ell }}c{{u}_{\tau }}}{{{{{q}''}}_{w}}} (9.167)

and using the dimensionless variables from eq. (9.158), the energy equation becomes

{{u}^{+}}\frac{\partial T}{\partial {{x}^{+}}}=\frac{\partial }{\partial {{y}^{+}}}\left( \frac{\alpha }{{{v}_{\ell }}}+\frac{{{\varepsilon }_{H}}}{{{v}_{\ell }}} \right)\frac{\partial {{T}^{+}}}{\partial {{y}^{+}}} (9.168)

The solution requires choosing a turbulence model for the specification of {{\varepsilon }_{M}}/\nu and of {{\varepsilon }_{H}}/{{\varepsilon }_{M}}. The detailed turbulence model and solution procedure can be found in Seban and Faghri (1976). The predicted result agrees very well with the experimental result of Chun and Seban (1971) for constant heat flux at wall. To predict local heat transfer in turbulent falling film flow on a wall heated under constant heat flux, the following empirical correlation, recommended by Chun and Seban (1971), can be used:

\frac{h}{{{k}_{\ell }}}{{\left[ \frac{\mu _{\ell }^{2}}{{{\rho }_{\ell }}\left( {{\rho }_{\ell }}-{{\rho }_{v}} \right)g} \right]}^{{1}/{3}\;}}=3.8\times {{10}^{-3}}{{\operatorname{Re}}^{0.4}}{{\Pr }^{0.65}} (9.169)

The average heat transfer coefficient can be obtained by substituting eq. (9.169) into eq. (9.112), i.e.,

\frac{\overset{\_}{\mathop{h}}\,}{{{k}_{\ell }}}{{\left[ \frac{\mu _{\ell }^{2}}{{{\rho }_{\ell }}\left( {{\rho }_{\ell }}-{{\rho }_{v}} \right)g} \right]}^{{1}/{3}\;}}=2.28\times {{10}^{-3}}\frac{\left( {{\operatorname{Re}}_{o}}-{{\operatorname{Re}}_{L}} \right)}{\left( \operatorname{Re}_{o}^{0.6}-\operatorname{Re}_{L}^{0.6} \right)}{{\Pr }^{0.65}} (9.170)

As was demonstrated in Example 9.4, eq. (9.170) is valid for cases with either constant wall temperature or constant heat flux. However, the methods for determining the Reynolds number at x = L will vary for different thermal boundary conditions on the wall. For cases with constant wall temperature, the Reynolds number at x = L, {{\operatorname{Re}}_{L}}, is obtained by substituting eq. (9.109) into eq. (9.170), i.e.,

\operatorname{Re}_{L}^{0.6}=\operatorname{Re}_{0}^{0.6}-9.12\times {{10}^{-3}}\frac{{{k}_{\ell }}L\left( {{T}_{w}}-{{T}_{v}} \right)}{{{\mu }_{\ell }}{{h}_{\ell v}}}{{\left[ \frac{{{\rho }_{\ell }}\left( {{\rho }_{\ell }}-{{\rho }_{v}} \right)g}{\mu _{\ell }^{2}} \right]}^{{1}/{3}\;}} (9.171)

For the case with constant heat flux, eq. (9.117) can be used to determine the Reynolds number at x = L, {{\operatorname{Re}}_{L}}. While gravitational force drives the falling film evaporation discussed here, liquid flow in a liquid film can also be driven by centrifugal force. Rahman and Faghri (1992) analyzed the processes of heating and evaporation in a thin liquid film adjacent to a horizontal disk rotating about a vertical axis at a constant angular velocity. The fluid emanates axisymmetrically from a source at the center of the disk and then is carried downstream by inertial and centrifugal forces. Closed-form analytical solutions were derived for fully developed flow and heat transfer. Simplified analyses were also presented for developing heat transfer in a fully-developed flow field. Rice et al. (2005) provided a detailed analysis for evaporation from a thin liquid film on a rotating disk including conjugate effects.