Evaporation from wavy laminar falling film

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Current revision as of 09:09, 7 July 2010

 Schematic of Nusselt evaporation.
Schematic of Nusselt evaporation.

Wavy flows of thin liquid films have higher heat transfer coefficients than smooth thin films. This effect is due to the former’s greater interfacial surface area and mixing action. Faghri and Seban (1985) analyzed a system in which a liquid film at an initial nondimensional temperature of zero flowed down a vertical wall and evaporated at the free liquid-vapor interface. The laminar liquid film thickness varied sinusoidally and the Reynolds number ranged between 35 and 472. The energy equation for a two-dimensional situation using the same vertical wall shown in the figure on the right is as follows:

\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\alpha \left( \frac{{{\partial }^{2}}T}{\partial {{x}^{2}}}+\frac{{{\partial }^{2}}T}{\partial {{y}^{2}}} \right)
(1)

The continuity equation for this system is

\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0
(2)

Faghri and Seban (1985) assumed a quasi-parabolic velocity profile in the x-direction as follows:

u=3\bar{u}(x,t)\left( \eta -\frac{{{\eta }^{2}}}{2} \right)
(3)

where the nondimensional coordinate η is defined as

\eta =\frac{y}{\delta }
(4)

and \bar{u} is the local mean velocity. Equation (2) can be rearranged and integrated with respect to y to obtain the velocity component v:

v=-\int\limits_{0}^{y}{\frac{\partial u}{\partial x}dy=-3}\left[ \frac{\partial \bar{u}}{\partial x}\delta \left( \frac{{{\eta }^{2}}}{2}-\frac{{{\eta }^{3}}}{6} \right)-\bar{u}\frac{\partial \delta }{\partial x}\left( \frac{{{\eta }^{2}}}{2}-\frac{{{\eta }^{3}}}{3} \right) \right]
(5)

It can be seen that eq. (3) has been substituted in order to find an expression for the v component of the velocity where the local mean velocity and film thickness δ are still unknown as functions of the wavy surface. To determine these two variables, the following mass balance can be written as a function of time and space:

\frac{\partial \delta }{\partial t}=-\frac{\partial }{\partial x}\int_{0}^{\delta }{udy=-\frac{\partial }{\partial x}\left( \bar{u}\delta  \right)}
(6)

The film thickness can be written in terms of its average value, \overline{\delta } , and a local amplitude φ.

\delta =\bar{\delta }(1+\varphi )
(7)

Assuming that the wave formation has a periodic characteristic with a velocity c, the following relations can be written for the fluctuation of local mean velocity and film thickness:

\frac{\partial \bar{u}}{\partial t}=-c\frac{\partial \bar{u}}{\partial x}
(8)
\frac{\partial \delta }{\partial t}=-c\frac{\partial \delta }{\partial x}
(9)

Combining eqs. (6) – (9) and integrating, the following expression is obtained:

\left( c-\bar{u} \right)\left( 1+\varphi  \right)=\left[ c-{{{\bar{u}}}_{0}} \right]
(10)

where {{\bar{u}}_{0}} is the average velocity for the average film thickness \bar{\delta }. For very small local amplitudes, φ < 1, \bar{u} may be approximated by expanding eq. (10) and neglecting the third-order terms.

\bar{u}={{\bar{u}}_{0}}+(c-{{\bar{u}}_{0}})\varphi -(c-{{\bar{u}}_{0}}){{\varphi }^{2}}
(11)

Substituting eqs. (7) and (11) into eq. (5), one obtains the velocity profile in terms of the wavy surface parameters:

\begin{align}
  & v=-3{{{\bar{u}}}_{0}}\bar{\delta }\frac{\partial \phi }{\partial x}\left\{ \left( \frac{c}{{{{\bar{u}}}_{0}}}-1 \right)(1-2\phi )\frac{\delta }{{\bar{\delta }}}\left( \frac{{{\eta }^{2}}}{2}-\frac{{{\eta }^{3}}}{6} \right) \right. \\ 
 & \left. \begin{matrix}
   {} & {}  \\
\end{matrix}-\left[ 1+\left( \frac{c}{{{{\bar{u}}}_{0}}}-1 \right)\phi -\left( \frac{c}{{{{\bar{u}}}_{0}}}-1 \right){{\phi }^{2}} \right]\left( \frac{{{\eta }^{2}}}{2}-\frac{{{\eta }^{3}}}{3} \right) \right\} \\ 
\end{align}
(12)

Assuming the wave is sinusoidal,

\varphi =A\sin \left[ \left( \frac{2\pi }{\lambda } \right)\left( x-ct \right) \right]
(13)

and introducing the nondimensional variable ξ defined as

\xi =\left( \frac{2\pi }{\lambda } \right)\left( x-ct \right)
(14)

where A is magnitude of the wave, and λ is the wavelength, eq. (1) is transformed from the (t, x, y) to the (ξ, η) coordinate system,

{{C}_{1}}\frac{\partial T}{\partial \xi }+{{C}_{2}}\frac{\partial T}{\partial \eta }={{C}_{3}}\frac{{{\partial }^{2}}T}{\partial {{\eta }^{2}}}+{{C}_{4}}\frac{{{\partial }^{2}}T}{\partial \eta \partial \xi }+{{C}_{5}}\frac{{{\partial }^{2}}T}{\partial {{\xi }^{2}}}
(15)

where the constants are as follows:

{{C}_{1}}=\frac{2\pi }{\lambda }(u-c)
(16)
\begin{align}
  & {{C}_{2}}=\frac{2\pi \eta }{\lambda }\frac{{\bar{\delta }}}{\delta }(c-u)A\cos \xi -{{\left( \frac{2\pi }{\lambda } \right)}^{2}}\frac{{\bar{\delta }}}{\delta }\alpha \eta A\sin \xi  \\ 
 & \begin{matrix}
   {} & {}  \\
\end{matrix}-2{{\left( \frac{2\pi }{\lambda } \right)}^{2}}\alpha \left( \frac{{\bar{\delta }}}{\delta } \right)\eta {{A}^{2}}{{\cos }^{2}}\xi +\frac{v}{\delta } \\ 
\end{align}
(17)
{{C}_{3}}=\frac{\alpha }{{{\delta }^{2}}}+\alpha {{\left( \frac{2\pi }{\lambda } \right)}^{2}}{{\left( \frac{{\bar{\delta }}}{\delta } \right)}^{2}}{{\eta }^{2}}{{A}^{2}}{{\cos }^{2}}\xi
(18)
{{C}_{4}}=-2\alpha {{\left( \frac{2\pi }{\lambda } \right)}^{2}}\eta \frac{{\bar{\delta }}}{\delta }A\cos \xi
(19)
{{C}_{5}}=\alpha {{\left( \frac{2\pi }{\lambda } \right)}^{2}}
(20)

These coefficients are evaluated with \bar{u} from eqs. (3) and (11), and v from eq. (12).

In the evaporating film with a fluid with a high latent heat of vaporization, the average film thickness over a wave length, \bar{\delta }, will not vary substantially with distance, x. Furthermore, the temporal average temperature profile will not vary with x far from where heating begins. Therefore, the boundary condition for eq. (15) that describes evaporation on a wavy film is approximated as below. The equality of temperatures at corresponding points in the period gives

T(0,η) = T(2π,η)
(21)

Boundary conditions for dimensionless temperature at the heated vertical wall and wavy surface are, respectively,

T(ξ,0) = 1
(22)
T(ξ,1) = 0
(23)

Therefore, the energy equation with the above boundary conditions can be solved to obtain the wavy condensate film’s temperature profile. With this profile, the heat flux at the wall can be found from

{{{q}''}_{w}}=-\frac{k}{\delta }{{\left. \frac{\partial T}{\partial \eta } \right|}_{\eta =0}}
(24)

and the heat flux normal to the wavy surface (η = 1) is as follows:

\frac{{{{\mathbf{{q}''}}}_{\delta }}}{k}=\left[ \frac{2\pi }{\lambda }\left( \frac{{\bar{\delta }}}{\delta }A\cos \xi  \right)\mathbf{i}-\frac{1}{\delta }\mathbf{j} \right]{{\left. \frac{\partial T}{\partial \eta } \right|}_{\eta =1}}
(25)

where \mathbf{i} and \mathbf{j} are unit vectors in the x- and y-directions, respectively. The average heat flow over a period 0 < ξ < 2π is found by integrating eqs. (24) and (25). For a sinusoidal wavy layer, such as the one considered here, this average is identical to the average obtained for the actual surface length. The average heat flux at the wall (η = 0) is approximated by

\frac{{{{\bar{{q}''}}}_{w}}}{k}=\frac{1}{2\pi }\int_{0}^{2\pi }{\frac{1}{\delta }{{\left. \frac{\partial T}{\partial \eta } \right|}_{0}}d\xi }
(26)

and the average normal heat flux at the wavy surface is as follows:

\frac{{{{\bar{{q}''}}}_{\delta }}}{k}=\frac{1}{2\pi }\int_{0}^{2\pi }{\frac{1}{\delta }{{\left. \frac{\partial T}{\partial \eta } \right|}_{\eta =1}}{{\left[ 1+{{\left( \frac{\partial \delta }{\partial x} \right)}^{2}} \right]}^{1/2}}d\xi }
(27)

For practical purposes, Chun and Seban (1971) suggested that the following correlation based on wavy laminar film condensation is valid for wavy laminar falling film evaporation:

{{h}_{x}}=0.876{{\left( \frac{\operatorname{Re}}{4} \right)}^{0.11}}{{h}_{Nusselt}}
(28)

An empirical correlation for the local heat transfer coefficient for laminar-wavy flow is obtained by combining eq. (28) with the following equation:

\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}}
(29)

and the result is

\frac{{{h}_{x}}}{{{k}_{\ell }}}{{\left[ \frac{\mu _{\ell }^{2}}{{{\rho }_{\ell }}\left( {{\rho }_{\ell }}-{{\rho }_{v}} \right)g} \right]}^{{1}/{3}\;}}=0.828{{\operatorname{Re}}^{-0.22}},\text{   }\operatorname{Re}>{{\operatorname{Re}}_{wavy}}
(30)

Chun and Seban (1971) suggested the following empirical correlation to predict the onset of wavy laminar flow:

\operatorname{Re}\ge {{\operatorname{Re}}_{wavy}}=2.43K{{a}^{-1/11}}
(31)

where Ka is Kapitza number, a ratio of surface tension to viscous force:

Ka=\frac{\mu _{\ell }^{4}g}{\left( {{\rho }_{\ell }}-{{\rho }_{v}} \right){{\sigma }^{3}}}
(32)

The average heat transfer coefficient is often relevant to the practical application and can be obtained by using the following equation

\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}}}
(33)

which is also valid for laminar flow with waves. Substituting eq. (31) into eq. (33), one obtains an empirical correlation of the average heat transfer coefficient as follows:

\frac{\overset{\_}{\mathop{h}}\,}{{{k}_{\ell }}}{{\left[ \frac{\mu _{\ell }^{2}}{{{\rho }_{\ell }}\left( {{\rho }_{\ell }}-{{\rho }_{v}} \right)g} \right]}^{{1}/{3}\;}}=\frac{\left( {{\operatorname{Re}}_{o}}-{{\operatorname{Re}}_{L}} \right)}{\left( \operatorname{Re}_{o}^{1.22}-\operatorname{Re}_{L}^{1.22} \right)}
(34)

Substituting the following equation

\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)}
(35)

into eq. (34), an equation correlating {{\operatorname{Re}}_{L}}, the Reynolds number at x = L, is obtained:

\operatorname{Re}_{L}^{1.22}=\operatorname{Re}_{0}^{1.22}-4\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}\;}}
(36)

References

Chun, K. R. and Seban, R. A., 1971, “Heat transfer to evaporating liquid films,” ASME Journal of Heat Transfer, Vol. 93, pp. 391-396.

Faghri, A. and Seban, R., 1985, “Heat Transfer in Wavy Liquid Films,” International Journal of Heat Mass Transfer, Vol. 28, pp. 506-508.

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.

Kutateladze, S.S., 1963, Fundamentals of Heat Transfer, Academic Press Inc., New York.

Kutateladze, S.S., 1982, “Semi-Empirical Theory of Film Condensation of Pure Vapors,” International Journal of Heat Mass Transfer, Vol. 25, pp. 653-660.