# Classical Nusselt Evaporation

The simplest evaporation heat transfer coefficient calculation uses the Nusselt treatment of falling film evaporation, as shown in the figure on the right. A flat plate with constant temperature Tw is placed vertically and a liquid film flows downward due to gravity. Evaporation takes place on the surface of the liquid film because the temperature of the vertical plate is above the saturation temperature of the vapor corresponding to the partial pressure of the vapor in the mixture. Since the heat must be transferred from the heat source (vertical wall) to the liquid-vapor interface, where evaporation takes place, this is an example of heterogeneous evaporation. The following assumptions are made:

1. The vapor is quiescent.

2. Inertia in the laminar liquid film is negligible.

3. Heat transfer through the liquid film is by conduction only, i.e., convective terms in the energy equation are negligible.

4. Constant wall temperature.

5. The process occurs at steady state.

6. A nonslip condition exists at the wall.

Proceeding from the assumptions of negligible inertia and constant fluid properties, the momentum balance is ${{\mu }_{\ell }}\frac{{{d}^{2}}{{u}_{\ell }}}{d{{y}^{2}}}=-\left( {{\rho }_{\ell }}-{{\rho }_{v}} \right)g$ (1)

Integrating eq. (1) twice, with respect to y and using the boundary conditions of non-slip at the wall (u=0 at y=0) and zero shear at the liquid-vapor interface ( $\partial u/\partial y=0$ at y = δ), the following velocity profile is obtained: ${{u}_{\ell }}=\frac{\left( {{\rho }_{\ell }}-{{\rho }_{v}} \right)g}{{{\mu }_{\ell }}}\left( \delta y-\frac{{{y}^{2}}}{2} \right)$ (2)

The mass flow rate per unit width is defined by integrating the velocity profile from the wall to the interface. $\Gamma =\int_{0}^{\delta }{{{\rho }_{\ell }}udy}$ (3)

The liquid velocity profile, eq. (2), is substituted into eq. (3) and the result is integrated from the wall to the interface. The result is then rearranged to obtain liquid film thickness: $\delta ={{\left[ \frac{3{{\mu }_{\ell }}\Gamma }{{{\rho }_{\ell }}\left( {{\rho }_{\ell }}-{{\rho }_{v}} \right)g} \right]}^{{1}/{3}\;}}$ (4)

Since heat transfer through the film is by conduction only, the heat transfer coefficient is ${{h}_{x}}={{k}_{\ell }}/\delta$. Considering eq. (4), the local heat transfer coefficient becomes ${{h}_{x}}={{k}_{\ell }}{{\left[ \frac{{{\rho }_{\ell }}\left( {{\rho }_{\ell }}-{{\rho }_{v}} \right)g}{3{{\mu }_{\ell }}\Gamma } \right]}^{{1}/{3}\;}}$ (5)

Note that the film thickness δ drops out of the equation, and the heat transfer coefficient h is now a function of the mass flow rate per unit width, Γ.

The local energy balance across the liquid film is ${{h}_{\ell v}}\frac{d\Gamma }{dx}=-{{h}_{x}}\left( {{T}_{w}}-{{T}_{v}} \right)$ (6)

Combining eqs. (5) and (6), one obtains, ${{\Gamma }^{{1}/{3}\;}}d\Gamma =-\frac{{{k}_{\ell }}}{{{h}_{\ell v}}}{{\left[ \frac{{{\rho }_{\ell }}\left( {{\rho }_{\ell }}-{{\rho }_{v}} \right)g}{3{{\mu }_{\ell }}} \right]}^{{1}/{3}\;}}\left( {{T}_{w}}-{{T}_{v}} \right)dx$ (7)

Integrating from the inlet to some arbitrary length L, $\int_{{{\Gamma }_{o}}}^{{{\Gamma }_{L}}}{{{\Gamma }^{{1}/{3}\;}}}d\Gamma =-\int_{0}^{L}{\frac{{{k}_{\ell }}}{{{h}_{\ell v}}}}{{\left[ \frac{{{\rho }_{\ell }}\left( {{\rho }_{\ell }}-{{\rho }_{v}} \right)g}{3{{\mu }_{\ell }}} \right]}^{{1}/{3}\;}}\left( {{T}_{w}}-{{T}_{v}} \right)dx$ (8)

A relation for Γ at outlet is obtained in terms of the inlet mass flow rate, Γ0, for the case of constant wall and vapor temperatures: $\int_{{{\Gamma }_{o}}}^{{{\Gamma }_{L}}}{{{\Gamma }^{{1}/{3}\;}}}d\Gamma =-\int_{0}^{L}{\frac{{{k}_{\ell }}}{{{h}_{\ell v}}}}{{\left[ \frac{{{\rho }_{\ell }}\left( {{\rho }_{\ell }}-{{\rho }_{v}} \right)g}{3{{\mu }_{\ell }}} \right]}^{{1}/{3}\;}}\left( {{T}_{w}}-{{T}_{v}} \right)dx$ (9)

Now the mass flow rate per unit width is related to the Reynolds number by $\operatorname{Re}=\frac{4\Gamma }{{{\mu }_{\ell }}}$ (10)

The local heat transfer coefficient from eq. (5) can be expressed in terms of Reynolds number as follows: $\frac{{{h}_{x}}}{{{k}_{\ell }}}{{\left[ \frac{\mu _{\ell }^{2}}{{{\rho }_{\ell }}\left( {{\rho }_{\ell }}-{{\rho }_{v}} \right)g} \right]}^{{1}/{3}\;}}={{\left( \frac{4}{3} \right)}^{{1}/{3}\;}}\frac{1}{{{\operatorname{Re}}^{{1}/{3}\;}}}=1.10{{\operatorname{Re}}^{-1/3}}$ (11)

Equation (11) for local heat transfer coefficient is valid for both laminar condensation and evaporation at both constant wall temperature and constant heat flux at wall.

To calculate the average heat transfer coefficient, $\bar{h}$, the energy flux over the entire length 0 < x < L is balanced with the mass flux from the interface: $\overset{\_}{\mathop{h}}\,=\frac{{{h}_{\ell v}}\left( {{\Gamma }_{0}}-{{\Gamma }_{L}} \right)}{L\left( {{T}_{w}}-{{T}_{v}} \right)}=\frac{{{\mu }_{\ell }}{{h}_{\ell v}}\left( {{\operatorname{Re}}_{0}}-{{\operatorname{Re}}_{L}} \right)}{4L\left( {{T}_{w}}-{{T}_{v}} \right)}$ (12)

Equation (12) is valid for laminar, wavy laminar, and turbulent with constant wall temperature.

Equation (6), which reflects the local energy balance across the liquid film, can be written in terms of Reynolds number as $\frac{d\operatorname{Re}}{{{h}_{x}}}=-\frac{4\left( {{T}_{w}}-{{T}_{v}} \right)}{{{\mu }_{\ell }}{{h}_{\ell v}}}dx$ (13)

Integrating eq. (13) in the interval of (0, L), one obtains $\int_{{{\operatorname{Re}}_{0}}}^{{{\operatorname{Re}}_{L}}}{\frac{d\operatorname{Re}}{{{h}_{x}}}}=-\frac{4\left( {{T}_{w}}-{{T}_{v}} \right)L}{{{\mu }_{\ell }}{{h}_{\ell v}}}$ (14)

Combining eqs. (12) and (14), the average heat transfer coefficient is expressed as $\overset{\_}{\mathop{h}}\,=-\frac{\left( {{\operatorname{Re}}_{0}}-{{\operatorname{Re}}_{L}} \right)}{\int_{{{\operatorname{Re}}_{0}}}^{{{\operatorname{Re}}_{L}}}{\frac{1}{{{h}_{x}}}d\operatorname{Re}}}$ (15)

Equation (15) for $\bar{h}$ is valid for laminar, wavy laminar, and turbulent flow.

Substituting eq. (11) into eq. (15), an expression for the average heat transfer coefficient for laminar flow in terms of Reynolds number is obtained: $\frac{\overset{\_}{\mathop{h}}\,}{{{k}_{\ell }}}{{\left[ \frac{\mu _{\ell }^{2}}{{{\rho }_{\ell }}\left( {{\rho }_{\ell }}-{{\rho }_{v}} \right)g} \right]}^{{1}/{3}\;}}={{\left( \frac{4}{3} \right)}^{{4}/{3}\;}}\frac{\left( {{\operatorname{Re}}_{o}}-{{\operatorname{Re}}_{L}} \right)}{\left( \operatorname{Re}_{o}^{{4}/{3}\;}-\operatorname{Re}_{L}^{{4}/{3}\;} \right)}$ (16)

In order to use eq. (16) to obtain the average heat transfer coefficient, it is necessary to know Reynolds number at x=L, ${{\operatorname{Re}}_{L}}$. It can be obtained by rewriting the relation for Γ, eq. (9), in terms of Reynolds number, as follows: $\operatorname{Re}_{L}^{{4}/{3}\;}=\operatorname{Re}_{o}^{{4}/{3}\;}-4{{\left( \frac{4}{3} \right)}^{{4}/{3}\;}}\frac{{{k}_{\ell }}\left( {{T}_{w}}-{{T}_{v}} \right)L}{\mu _{\ell }^{{4}/{3}\;}{{h}_{\ell v}}}{{\left[ \frac{{{\rho }_{\ell }}\left( {{\rho }_{\ell }}-{{\rho }_{v}} \right)g}{3{{\mu }_{\ell }}} \right]}^{{1}/{3}\;}}$ (17)

The above analysis is valid for falling film evaporation on a vertical wall maintained at a constant temperature.

## 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.